ON THE EXPLICIT RECONSTRUCTION OF A RIEMANN SURFACE FROM ITS DIRICHLET-NEUMANN OPERATOR

GENNADI HENKIN AND VINCENT MICHEL

Abstract. This article gives a complex analysis lighting on the problem which consists in restoring a bordered connected riemaniann surface from its boundary and its Dirichlet-Neumann operator. The three aspects of this problem, unicity, reconstruction and characterization are approached.

Contents

1. Statements of the main results 1 2. Intrinsic EIT on Riemann surfaces 11 3. Unicity under existence assumption 14 3.1. Proof of theorem 1. 16 4. Existence and reconstruction, proof of theorem 2 17 5. Proofs of characterizations theorem 3a, 3b and 3c 19 5.1. Proof of A.theorem 3a 20 5.2. Proof of B.theorem 3a 21 5.3. Proof of C.theorem 3a 22 5.4. Proof of A.theorem 3c 23 5.5. Proof of B.theorem 3c 24 5.6. Proof of theorem 3b. 27 6. Characterizations, effective or affine 27 6.1. Explicit integro-differential characterization 27 6.2. Affine characterization 32 References 32

1. Statements of the main results

Let X be an open bordered riemannian real surface (i.e. the interior of an oriented riemannian two dimensional real manifold whose all components have non trivial one dimensional smooth boundary) and g its metric. Using the boundary control method, Belishev and Kurylev [4][6] have started the study of the inverse problem consisting in recovering (X , g) from the operators N_{λ} : C^{∞} (bX) ∋ u ↦ (∂ũ_{λ}/∂ν)_{bX} where bX is the boundary of X, ν is the normal exterior unit to bX and ũ_{λ} is the unique solution of Δ_{g}U = λU such that U |_{bX} = u. The principal result of [6] implies that the knowledge of λ ↦ N_{λ} on an non empty open set of ℝ_{+} determines (X , g) up to isometry. The important question to know if (X , g) is uniquely determined by only one operator N_{λ_{∗}} with λ_{∗} ≠ 0 remains open. This article mainly deals with the case of the Dirichlet-Neumann operator N_{X} := N_{0}. Section 2 gives an intrinsic interpretation Electrical Impedance Tomography on manifolds, EIT for short, in terms of Inverse-Dirichlet-Neumann problem for twisted Laplacian. In dimension two, this clearly underline how the complex structure of Riemannian surfaces is involved.

Date: September 29, 2005.

1991 Mathematics Subject Classification. Primary 58J32, 32C25; Secondary 35R30 .

Two surfaces in the same conform class which have the same oriented boundary and whose metrics coincides there, need to have the same Dirichlet-Neumann operator. Conversely, Lassas and Uhlmann [21] have proved for a connected X that the conform class and so the complex structure of (X , g) is determined by N_{X}. Hence, it is relevant to consider X as a Riemann surface. In [5], using also the full knowledge of N_{X}, Belishev gives another proof of the above unicity by recovering abstractly X as the spectre of the algebra of boundary values of functions holomorphic on X and continuous on \overline{X} = X ∪ bX. It turns out in our theorems 1 and 2 that only three generic functions on the boundary and their images by N_{X} are sufficient for unicity to hold and to reconstruct X by integral Cauchy type formulas. Theorems 3a, 3b and 3c deal with characterizations of data of the type (bX , N_{X}) where X is a Riemann surface.

While the frame of bordered manifolds is sufficient for real analytic boundaries, characterization statements lead to consider a wider class of manifolds. In this article, ( \overline{X} , γ ) is a Riemann surface with almost smooth boundary if the following holds^{(}^{1}^{)} : \overline{X} is a compact metrizable topological manifold which is the closure of X = \overline{X} \γ, X is a Riemann surface^{(}^{2}^{)}, h^{2} ( \overline{X} ) < ∞, γ is a smooth real curve^{(}^{3}^{)} and the set \overline{X}_{sing} of points of γ where \overline{X} has not smooth boundary satisfies h^{1} (\overline{X}_{sing}) = 0 ; \overline{X} \\overline{X}_{sing} is denoted \overline{X}_{reg}.

If ( \overline{X} , γ ) is a Riemann surface with almost smooth boundary, classical results contained in [1] implies a Riemann’s existence theorem : a real valued function u of class C^{1} on γ has a unique continuous extension ũ to \overline{X} which is harmonic on X, smooth on \overline{X}_{reg} and satisfy ∫_{X} i ∂ũ∧\overline{∂}ũ < +∞. Moreover, N_{X} u still make sense as the element of the dual space of C^{1} (γ) which equals ∂ũ/∂ν on γ\\overline{X}_{sing} (see proposition 12).

In the sequel, γ is a smooth compact oriented real curve without component reduced to a point, N is an operator from C^{1} (γ) to the space of currents on γ of degree 0 and order 1 (i.e. functionals on C^{1} 1-forms on γ), τ is a smooth generating section of T γ and ν is another vector field along γ such that the bundle T generated by (ν_{x}, τ_{x}), x ∈ γ, has rank 2 ; γ is assumed to be oriented by τ and T by (ν, τ).

The inverse Dirichlet-Neumann problem for (γ, N, T ) is to find, when it exists, an open riemaniann surface (X , g) with almost smooth boundary γ such that for all x ∈ γ ∩ \overline{X}_{reg}, (ν_{x}, τ_{x}) is a positively oriented orthonormal basis of T_{x}\overline{X} and for all u ∈ C^{1} (γ), Nu = N_{X} u in the sense of currents. As these conditions do not distinguished between metrics g in a same conformal class, we look after X as a Riemann surface. The connection between real and complex analysis in the IDN-problem is realized through the operators L and θ defined for u ∈ C^{1} (γ) by

1A Stokes formula holds automatically for such manifolds (see lemma 11 in section 3).

2h^{d} is the d-dimensionnial Hausdorff measure.

3It could have been possible to allow singularities on γ itself but we have avoid it for the sake of simplicity of statements. Likewise, we consider only smooth DN-datas in the sequel.

(1.1)  Lu = \frac{1}{2} (Nu − i T u) & θu = (Lu) (ν^{∗} + iτ^{∗})

where T is the tangential derivation by τ and (ν_{x}^{∗}, τ_{x}^{∗}) is the dual basis of (ν_{x}, τ_{x}) for every x ∈ γ. Note that in the sense of currents, the equality Nu = N_{X} u is equivalent to the identity ∂ũ = θu, the tilde denoting, as through all this article, continuous harmonic extension to X

If (\overline{X} , γ) is a Riemann surface with almost smooth boundary, g is a hermitian metric on X for which (τ_{x}, ν_{x}) is a positively oriented orthonormal basis of T_{x}^{∗}X for x ∈ γ outside σ = \overline{X}_{sing} and if ρ ∈ C^{0} (\overline{X} ) ∩ C^{∞} (\overline{X} \σ) is a defining function of γ in \overline{X}, then (ν^{∗}, τ^{∗}) = \frac{1}{|dρ|_{g}} (dρ, d^{c}ρ) on γ\σ where

d^{c} = i( \overline{∂} − ∂) and ∂ũ = (Lu) |∂ρ|_{g}^{−1} ∂ρ = θu on γ\σ for all u ∈ C^{1} (γ).

Main hypothesis. In addition to the assumptions on γ, we consider in all this paper u_{0}, u_{1}, u_{2} ∈ C^{∞} (γ) three real valued functions only ruled by the main hypothesis that

(1.2)  f = (f_{1}, f_{2}) = ( (Lu_{ℓ}) / (Lu_{0}) )_{ℓ=1,2} = ( (θu_{ℓ}) /(θu_{0}) )_{ℓ=1,2}

is an embedding of γ in ℂ^{2} considered as the complement of {w_{0} = 0} in the complex projective plane ℂℙ_{2} with homogeneous coordinates (w_{0} : w_{1} : w_{2}). Prop. 0 whose proof is omitted shows this is somehow generic :

Proposition 0. Assume γ, u_{0}, u_{1} real analytic and that f_{1} is non constant on each connected component of γ. For any function u_{2} ∈ C^{ω} (γ), one can construct v_{2} ∈ C^{ω} (γ), arbitrarily close to u_{2} in C^{2} norm, such that (f_{1}, (Lv_{2}) / (Lu_{0}) ) is an embedding of γ into ℂ^{2}.

Assuming that u = (u_{ℓ})_{0⩽ℓ⩽2} satisfies the main hypothesis, we set θu = (θu_{ℓ})_{0⩽ℓ⩽2} and call (γ, u, θu) a restricted DN-datum for an open Riemann surface X if X has almost smooth boundary γ, (∂ũ_{ℓ}) |_{γ\σ} = θu_{ℓ} for 0 ⩽ ℓ ⩽ 2 and the well defined meromorphic quotient F_{ℓ} of (1,0)-forms (∂ũ_{ℓ}) / (∂ũ_{0}) extends f_{ℓ} to X in the sense that for every x_{0} ∈ γ, lim_{x→x_{0}, x∈X}F (x) exists

and equals f (x_{0}). If γ and f are real analytic, this last property holds automatically.

We define an isomorphism between two Riemann surfaces with almost smooth boundary ( \overline{X} , γ ) and ( \overline{X′}, γ′ ) as a map from \overline{X} to \overline{X′} which realizes a complex analytic isomorphism between X and X′. As the definition of a Riemann surface with almost smooth boundary implies that its boundary is locally a Jordan curve in its double which is the compact Riemann surface obtained by gluing along its boundary its conjugate (see [1]), a theorem of Caratheodory implies that if Φ : X → X′ is a complex analytic isomorphism, Φ and Φ^{−1} extend continuously to γ and γ′ so that Φ becomes a homeomorphism from \overline{X} to \overline{X′}. Hence, Φ is a diffeomorphism between manifolds with boundary from \overline{X}_{reg} ∩ Φ^{−1} ( \overline{X′}_{reg} ) to \overline{X′}_{reg} ∩ Φ (\overline{X}_{reg}).

The first theorem of this article is a significative improvement of results in [6][21] on how unique X can be when a restricted DN-datum is specified. Theorem 1. Assume that X and X′ are open Riemann surfaces with restricted DN-datum (γ, u, θu). Then, there is an isomorphism of Riemann surfaces with almost smooth boundary between X ∪ γ and X′ ∪ γ whose restriction on γ is the identity. Remarks. 1. If E ⊂ γ and h^{1} (E ∩ c) > 0 for each connected component c of γ, meromorphic functions are uniquely determined by their values on E and it follows that th. 1 conclusions hold when N_{X′}u_{ℓ} = N_{X} u_{ℓ} is ensured only on E and the meromorphic functions (∂ũ_{ℓ}) / (∂ũ_{0}) are continuous near γ. This includes [21, th. 1.1.i] which is stated for a connected X.

2. The proof of theorem 1 also contains the fact that two connected compact Riemann surfaces Z and Z′ are isomorphic when they share a same real smooth curve γ which can be embedded into ℂ^{2} by a map which extends meromorphically both to Z and Z′ and continuously near γ.

The assumption on u_{0}, u_{1} and u_{2} is used only to ensure that the map f defined by (1.2) is an embedding of γ into ℂ^{2} extending meromorphically to X into F = (∂ũ_{ℓ}/∂ũ_{0})_{ℓ=1,2}. Moreover, theorem 10 in section 3 implies that if X has almost smooth boundary and solves the IDN-problem, the map F enable to see \overline{X} as a normalization of the closure of a complex curve^{(}^{4}^{)} of ℂℙ_{2}\f (γ) uniquely determined by γ. This shows that in each characterization theorem 3a, 3b, 3c, the constructed Riemann surface is, up to isomorphism, the only one which has a chance to solve the IDN-problem.

Our next result explains how to recover F (X) and ∂ũ_{ℓ} from θu_{ℓ} and the intersection of F (X) with the lines Δ_{ξ} = \{z_{2} := \frac{w_{2}}{w_{0}} = ξ\}, ξ ∈ ℂ. Desingularization arguments enable then the reconstruction of X from F (X). Theorem 2. If X is an open Riemann surface with restricted DN-datum (γ, u, θu), the following holds :

1) The map f defined by (1.2) has a meromorphic extension F to X and there are discrete sets A and B in X and Y = F (X) \f (γ) respectively such that F : X \A → Y\B is one to one.

2) Almost all ξ_{∗} ∈ ℂ has a neighborhood W_{ξ_{∗}} such that for all ξ in W_{ξ_{∗}}, Y_{ξ} = Y ∩ Δ_{ξ} = ∪_{1⩽j⩽p}{(h_{j} (ξ) , ξ)} where h_{1}, ..., h_{p} are p mutually distinct

holomorphic functions on W_{ξ_{∗}} whose symmetric functions S_{h,m} = Σ_{1⩽j⩽p}h_{j}^{m}

are recovered by the Cauchy type integral formulas

(E_{m,ξ})  \frac{1}{2πi} ∫_{γ} \frac{f_{1}^{m}}{f_{2} − ξ} df_{2} = S_{h,m} (ξ) + P_{m} (ξ) , m ∈ ℕ,

where P_{m} is a polynomial of degree at most m. More precisely, the system E_{ξ} = (E_{m,ξ_{ν}})\begin{matrix}0⩽m⩽B−1 \\ o⩽ν⩽A−1 \end{matrix} enables explicit computation of h_{j} (ξ_{ν}) and P_{m} if

A ⩾ B ⩾ 2p + 1 and ξ_{0}, ..., ξ_{A} are mutually distinct points.

3) For almost all ξ_{∗} ∈ ℂ, W_{ξ_{∗}} can be chosen so that B ∩ ∪_{ξ∈W_{ξ_{∗}}}Y_{ξ} = ∅ and

∂ũ_{ℓ}, 0 ⩽ ℓ ⩽ 2, can be reconstructed in F^{−1}( ∪_{ξ∈W_{ξ_{∗}}}Y_{ξ}) from the well defined meromorphic quotient (∂ũ_{ℓ}) / (∂F_{2}) thanks to the Cauchy type formulas

4That is a pure 1-dimensionnal complex analytic subset of an open subset of ℂℙ_{2}.

(T_{m,ξ}) \frac{1}{2πi} ∫_{γ} \frac{f_{1}^{m}}{f_{2} − ξ} θu_{ℓ} = ∑_{1⩽j⩽p} h_{j} (ξ)^{m} \frac{∂ũ_{ℓ}}{∂F_{2}} (F^{−1} (h_{j} (ξ) , ξ)) +Q_{m} (ξ)

where m is any integer and Q is a polynomial of degree at most m.

Remark. The number α of connected components of X can be computed by the following algorithm : let γ_{1} be a component of γ and let λ_{1} be a function which is zero on γ_{j} for j ≠ 0 and non constant on γ_{1} ; then if X_{1} is the component of X whose boundary contains γ_{1}, Nλ_{1} ≠ 0 on each component γ_{1}, ..., γ_{k} of γ which with γ_{1} are the components of bX_{1}. Iterating this with components γ different from γ_{1}, ..., γ_{k}, yields a process with α steps.

The numerical resolution of (E_{ξ}) and the study of its stability requires an estimate of the number I_{Δ_{ξ}} of points of intersection, multiplicities taken in account, of Y with Δ_{ξ}. To achieve this, it is sufficient to estimates the number I_{Δ} of intersection points of Y with a ℂℙ_{2}-line Δ generic in the sense that Δ does not contained the germ of a component of Y near γ. Indeed, if L (resp. L_{ξ}) denotes a linear homogeneous form defining Δ (resp. Δ_{ξ}),

I_{Δ_{ξ}} − I_{Δ} = \frac{1}{2πi} ∫_{γ} (L_{ξ}/L)^{−1} d (L_{ξ}/L) .

Thus, an a priori upper bound of I_{Δ} for any particular line Δ would be very useful. This open problem is related, because of the Ahlfors theorem on covering surface, to the computation of the genus g_{X} of X from some DNdatum when X is connected. Under the condition γ is connected, Belishev [5] has shown that 2g_{X} is the rank of Id + (N_{X} T^{−1})^{2} acting on the space of smooth functions on γ admitting a smooth primitive, T^{−1} being a primitive operator. Yet, a formula for g_{X} involving only the action of N_{X} on a finite generic set of functions has to be found.

The third aspect of the IDN-problem, characterization of what can and should be a DN-datum, have lead us to allow X to have only almost smooth boundary. Th. 3a below explicitly characterizes the only right candidate for X while its part C gives a test which discriminates which (γ, u, θu) are DN-data and which are not. To perform it, we need a Green function for \overline{X} relatively to a domain D of Z containing \overline{X}, that is a smooth symmetric function g defined on D × D without its diagonal such that each g (., z) is harmonic on D\ \{z\} and has singularity \frac{1}{2π} ln dist (., z) at z, the distance being computed in any hermitian metric on Z.

Theorem 3a. Assume that the main hypothesis is valid and consider

(1.3)  G : ℂ^{2} ∋ (ξ_{0}, ξ_{1}) ↦ \frac{1}{2πi} ∫_{γ} f_{1} \frac{d (ξ_{0} + ξ_{1}f_{1} + f_{2})}{ξ_{0} + ξ_{1}f_{1} + f_{2}}.

A. If an open Riemann surface X has restricted DN-datum (γ, u, θu), then almost all point ξ_{∗} of ℂ^{2} has a neighborhood where one can find mutually distinct holomorphic functions h_{1}, ..., h_{p} such that

0 = \frac{∂^{2}}{∂ξ_{0}^{2}} (G − ∑_{1⩽j⩽p} h_{j}) (1.4)

h_{j} \frac{∂h_{j}}{∂ξ_{0}} = \frac{∂h_{j}}{∂ξ_{1}} , 1 ⩽ j ⩽ p. (1.5)

B. Conversely, assume γ is connected and the conclusion of A is satisfied in a connected neighborhood W_{ξ_{∗}} of one point (ξ_{0∗}, ξ_{1∗}). Then, if (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0, there is an open Riemann surface X with almost smooth

boundary^{(}^{5}^{)} γ where f extends meromorphically. If (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0, the same conclusion holds for a suitable orientation of γ.

C. Assume that ( \overline{X} , γ ) is a Riemann surface with almost smooth boundary. Let Z be the double of X, D a smooth domain of Z containing \overline{X} and g a Green function for \overline{X} relatively to D. Then, (γ, u, θu) is actually a restricted DN-datum if and only if for any z ∈ D\\overline{X},

(1.6)  ∫_{γ} u_{ℓ} (ζ) ∂_{ζ}g (ζ, z) + g (ζ, z) \overline{θu_{ℓ} (ζ)} = 0, 0 ⩽ ℓ ⩽ 2.

Remarks. 1. The connectness of γ is essentially used to ensure that any possible solution to the IDN-problem has to be connected. Taking in account the remark following th. 2, one may weaken the connectness assumption on γ into the requirement that the given DN-datum ensures that possible solutions are connected. Then, the conclusions of th. 3a.B are still true (see the proof).

2. The proof includes that if γ and f are real analytic, ( \overline{X} , γ ) is a manifold with boundary in the classical sense.

3. Emphasizing on f_{2} instead of f_{1}, one can consider

G_{2} : ξ ↦ \frac{1}{2πi} ∫_{γ} f_{2} \frac{d (ξ_{0} + ξ_{1}f_{1} + f_{2})}{ξ_{0} + ξ_{1}f_{1} + f_{2}}.

If h_{j} is linked to h_{j,2} by 0 = ξ_{0} + ξ_{1}h_{j} + h_{j,2}, (h_{1}, ..., h_{p}) satisfy (1.5) and

(1.4) if and only \frac{∂^{2}}{∂ξ_{0}^{2}} (G_{2} − Σ_{1⩽j⩽p}h_{j,2}) = 0 and h_{j} \frac{∂h_{j,2}}{∂ξ_{0}} = \frac{∂h_{j,2}}{∂ξ_{1}} , 1 ⩽ j ⩽ p.

4. Select H = {h_{1}, ..., h_{p}} satisfying (1.5) and minimal for (1.4). Then, section 5.2 and proposition 14 shows that there is τ′ ⊂ δ = f (γ) such that h^{1} (τ′) = 0 and X is a normalization of the abstract curve Y ∪ τ′ where, when H = ∅, Y is the polynomial hull of δ in the affine complex plane

ℂ_{ξ_{∗}}^{2} = {w ∈ ℂℙ^{2} ; ξ_{∗}w = ξ_{0∗}w_{0} + ξ_{1∗}w_{1} + w_{2} ≠ 0} .

and, otherwise, Y is the analytic extension in ℂℙ_{2}\δ of the union of the graphs of the functions (1 : h_{j} : −ξ_{0} − ξ_{1}h_{j}), 1 ⩽ j ⩽ p. Hence, when H is minimal, decomposition (1.4) of G is unique up to order and Card H is the minimal number p for which such a decomposition exists. Moreover, theorem 10 implies that the only Riemann surfaces X which has a chance to solve the IDN-problem are normalizations of Y.

5With [18, example 10.5], one can construct smooth restricted DN-datas for which the solution of the IDN-problem is a manifold with only almost smooth boundary.

The vanishing of ∂^{2}G/∂ξ_{0}^{2} in a connected neighborhood W_{ξ_{∗}} of ξ_{∗} ∈ ℂ^{2} is known to be equivalent to the fact that δ = f (γ) satisfy the classical Wermer-Harvey-Lawson moment condition in^{(}^{6}^{)} ℂ_{ξ_{∗}}^{2} : for all k_{1}, k_{2} ∈ ℕ, ∫_{δ} z_{1}^{k_{1}}z_{2}^{k_{2}}dz_{2} = 0 where z = (w_{j}/ξ_{∗}w)_{j=1,2} (see [12, cor. 1.6.2]). It is proved in [29] for the real analytic case and in [7][18] for the smooth case that for a suitable orientation of γ, this moment condition guarantees the existence in ℂ_{ξ_{∗}}^{2}\δ of a unique complex curve Y with finite mass and boundary ±δ in the sense of currents. In [2], Alexander and Wermer have improved this Wermer-Bishop-Harvey-Lawson statement by showing that a closed oriented smooth connected real curve δ of ℂ^{2} is, with its given orientation, the boundary, in the sense of currents, of a complex curve of finite mass in ℂ_{ξ_{∗}}^{2}\δ if and only if \frac{1}{2πi} ∫_{δ} \frac{dA}{A} ⩾ 0 for any polynomial A which does not vanish on γ. Hence, in case (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0 it is sufficient to find one polynomial A such that

∫_{f(γ)} \frac{dA}{A} ≠ 0 to determine the correct orientation of γ.

Note that the case (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0 occurs only for very special DN-

data since it implies that for ℓ = 1, 2, f_{ℓ} admits a ℂ_{ξ_{∗}}^{2}-valued holomorphic extension to X. Proposition 20 proposes another result of this kind for some other special DN-data when they are available.

To palliate the difficulty of computing Green functions, the theorem below proposes another way to achieve the same goals : select the right candidates for X and extension of θu_{ℓ}; check this yields a solution.

Theorem 3b. Assume that the main hypothesis is valid. Let G be the function defined by (1.3) and let be G̃ the form which in ℂℙ_{2} with homogenous coordinates η = (η_{0} : η_{1} : η_{2}) is given by

(1.7)  G̃ = ∑_{0⩽ℓ⩽2} \frac{1}{2πi} (∫_{γ} \frac{θu_{ℓ}}{η_{0} + η_{1}f_{1} + η_{2}f_{2}} ) dη_{ℓ} = ∑_{0⩽ℓ⩽2} G̃_{ℓ} dη_{ℓ}

A. If an open Riemann surface X has restricted DN-datum (γ, u, θu), then, a1) Almost all points η_{∗} = (ξ_{∗0} : ξ_{∗1} : 1) of ℂℙ_{2} has a neighborhood where G̃ can be written as the sum of p holomorphic closed forms g_{j} = Σ_{0⩽ℓ⩽2}g_{j,ℓ} dη_{ℓ}

such that (h_{j}) = (g_{j,1}/g_{j,0})_{1⩽j⩽p} satisfy (1.5). a2) The form Θ_{ℓ} = ∂ũ_{ℓ}, 0 ⩽ ℓ ⩽ 2, satisfies

(1.8)  ∫_{c} Re Θ_{ℓ} = 0

for all c in the first homology group H_{1} (X) of X.

B. b1) Assume γ is connected and there is η_{∗} = (ξ_{∗0} : ξ_{∗1} : 1) and a connected neighborhood W_{ξ_{∗}} of ξ_{∗} = (ξ_{∗0}, ξ_{∗1}) such that (a1) is true for all η ∈ W_{η_{∗}} = {(ξ_{0} : ξ_{1} : 1) ; (ξ_{0}, ξ_{1}) ∈ W_{ξ_{∗}}}. Then, there exists an open Riemann surface X, topologically bordered by γ, where f extends meromorphically and each θu_{ℓ} extends weakly^{(}^{7}^{)} into a meromorphic (1,0)-form Θ_{ℓ} outside a set Σ of zero length^{(}^{8}^{)}. b2) In addition to (a1), assume that Θ_{ℓ} satisfies ∫ i Θ_{ℓ} ∧ \overline{Θ_{ℓ}} < +∞ and (1.8). Then if (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0, ( \overline{X} , γ ) is a manifold with almost smooth

6When ξ_{∗} belongs to the connected component of infinity of {ξ ∈ ℂ^{2} ; ∀w ∈ δ, ξw ≠ 0}, this moment condition is equivalent to the moment condition in ℂ_{(1,0)}^{2} and also to the vanishing of G on this component.

boundary ; the same conclusion holds when G̃ |_{W_{η_{∗}}} = 0 if γ has a suitable orientation. If (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0 but G̃ |_{W_{η_{∗}}} ≠ 0, then either X is a domain with boundary γ in a normalization of an algebraic curve of ℂℙ_{2}, either \overline{X} is a compact Riemann surface where γ is a slit^{(}^{9}^{)}. In all cases, u_{ℓ} admits a continuous extension ũ_{ℓ} to \overline{X} which is harmonic in X and such that Θ_{ℓ} = ∂ũ_{ℓ}, which means that Nu_{ℓ} is actually the DN-datum of X for u_{ℓ}.

Remarks. 1. When (a1) holds, h_{j,2} = \frac{g_{j,2}}{g_{j,0}} verify h_{j} \frac{∂h_{j,2}}{∂ξ_{0}} = \frac{∂h_{j,2}}{∂ξ_{1}} , 1 ⩽ j ⩽ p.

2. Formulas (E_{m,ξ}) and (T_{m,ξ}) enable direct reconstruction of a projective presentation of X and forms Θ_{ℓ}.

Theorem 3b is obtained by a normalization of a singular version of the IDN-problem which is more explicit. When X is smooth, the harmonicity of a distribution U is equivalent to the fact that ∂U is holomorphic. For the case where X is a complex curve of an open set in ℂℙ_{2}, we need two of the several non equivalent definitions of holomorphic (1,0)-forms.

At first, we use the weakly holomorphic forms introduced by Rosenlicht [25] which can be defined as meromorphic (1,0)-forms ψ such that ψ ∧ [X] is \overline{∂}-closed current of ℂℙ_{2}. Such forms ψ are also characterized by the fact that p_{∗}ψ is a usual holomorphic (1, 0)-form for any holomorphic proper function p : X → ℂ. A distribution U is defined as weakly harmonic if ∂U is weakly holomorphic.

Assume now X lies in ℂℙ_{2} and that X is bounded in the sense of currents by γ. A distribution U on X is said almost smooth up to the boundary if it is the case near each p ∈ γ where ( \overline{X} , γ ) is a manifold with boundary and if U has a restriction on γ in the sense of currents. When u is a smooth function on γ, a weakly harmonic extension of u to X is a weakly harmonic distribution U almost smooth up to boundary whose restriction on γ is u. Since two weakly harmonic extension U_{1} and U_{2} of u to X are equal when ∂U_{1} = ∂U_{2} on γ in the sense of currents, we consider a weak Cauchy-Dirichlet problem : a data is a smooth function u on γ and a smooth section λ of T_{γ}^{∗}X ; a solution is a weakly harmonic function U almost smooth up to γ such that u = U |_{γ} and λ = (∂U)_{γ} in the sense of currents ; when it exists, such an U is unique and is denoted ũ as any harmonic extension in this article. In connection with this notion, we define a weak restricted data as a triplet ( γ, u, θu) where u = (u_{ℓ})_{0⩽ℓ⩽2} (resp. θu = (θu_{ℓ})_{0⩽ℓ⩽2}) is a triplet of smooth functions (resp. (1,0)-forms) on γ such that θu_{ℓ} = (∂ũ_{ℓ})_{γ} in the sense of currents.

7It means that ∫_{γ} φθu_{ℓ} = ∫_{X} d (φΘ_{ℓ}) = ∫_{X} ( \overline{∂}φ ) ∧ Θ_{ℓ} holds for any Lipschitz function φ on \overline{X} which is a holomorphic function of f near points of Σ and singular points of ( \overline{X} , γ ) ; if ( \overline{X} , γ ) is a manifold with boundary, this definition means that Θ_{ℓ} |_{γ} = θu_{ℓ} in the usual sense.

8Basing on [11, example 1], one can construct examples where (a1) is satisfied while the weak extension Θ_{ℓ} has essential singularities on some zero length set Σ.

9This means that \overline{X} \γ is connected.

The weak CD-problem has its own interest and arise naturally in the proof of th. 3b. However, the original IDN-problem requires a more restrictive notion of harmonicity. According to Griffiths [15], holomorphic forms (resp. harmonic functions) are by definition push forward of holomorphic forms (resp. harmonic functions) on a normalization of X. Equivalently, a real function U on X is harmonic if and only if U is harmonic in the regular part X_{reg} of X and ∫_{X_{reg}} i ∂U ∧ \overline{∂}U < +∞. This notion is close in spirit to a Riemann characterization of the harmonic function with given boundary value u as the smooth function extending u to \overline{X} and minimizing the preceding integral. We can now state a singular version of th. 3b.

Theorem 3c. Consider in ℂℙ_{2}\ {w_{0} = 0} a smooth oriented real curve γ, three functions u_{0}, u_{1}, u_{2} in C^{∞} (γ) and θ_{0}, θ_{1}, θ_{2} three smooth sections of Λ^{1,0}T_{γ}^{∗}ℂℙ_{2} such that du_{ℓ} = 2 Re θ_{ℓ}, 0 ⩽ ℓ ⩽ 2 and linked by the relations θ_{1} = z_{1}θ_{0}, θ_{2} = z_{2}θ_{0}. Let G and G̃ be the form given by (1.3) and (1.7) but with (f_{1}, f_{2}) = (z_{1}, z_{2}).

A. Assume γ bounds, in the sense of currents, a complex curve X of ℂℙ_{2}\γ which has finite volume and weak restricted DN-datum (γ, u, θu). Then, a1) The conclusions of theorem 3b.A.a1 are valid. a2) The form Θ_{ℓ} = ∂ũ_{ℓ} satisfies (1.8) for all c in H_{1} (X_{reg}).

B. b1) Conversely, assume that γ is connected and that (a1) is valid for one point η_{∗} = (ξ_{0∗} : ξ_{1∗} : 1). Then, there is in ℂℙ_{2}\γ a complex curve X of finite mass where each θ_{ℓ} extends weakly on X into a weakly holomorphic (1, 0)-form Θ_{ℓ}. Moreover, if (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0, then X has boundary γ

in the sense of currents ; the same conclusion holds if G̃ |_{W_{η_{∗}}} = 0 but for a suitable orientation of γ. If (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0 but G̃ |_{W_{η_{∗}}} ≠ 0, either X is a domain in an algebraic curve of ℂℙ_{2} and has boundary γ in the sense of currents, either \overline{X} itself is an algebraic curve of ℂℙ_{2} where γ is a slit.

b2) If in addition (1.8) is satisfied by Θ_{ℓ} for all c ∈ H_{1} (X_{reg}), then u_{ℓ} has a (unique) weakly harmonic extension ũ_{ℓ} and Θ_{ℓ} = ∂ũ_{ℓ}. If Θ_{ℓ} also satisfy ∫_{X_{reg}} i Θ_{ℓ} ∧ \overline{Θ_{ℓ}} < +∞, then ũ_{ℓ} is harmonic.

Remark. It is possible that X has zero boundary in the sense of currents. This occurs only in the exceptional case where \overline{X} is a compact complex curve of ℂℙ_{2} and (so is algebraic) where γ is a slit. In the other cases, X has boundary ±γ in the sense of currents and a result of Chirka [10] gives that outside a zero one Hausdorff dimensional subset, ( \overline{X} , ±γ ) is locally a manifold with boundary.

The proofs of the preceding theorems are given in sections three to five. They use the results on the complex Plateau problem started in [29][7], developed in [18][17][11] for ℂ^{n} and in [20][12][19] for ℂℙ_{n}.

The non constructive existence criterions of theorems 3a, 3b and 3c may incite to seek a less general but more effective characterization. It has been already mentioned after theorem 3a that in the special case p = 0, the condition (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0 together with the Alexander-Wermer moment criterion gives an effective tool but only when special DN-data are at hand.

For p > 0, the main result of [12] is that conditions of type (1.4) and (1.5) characterize the fact that a given closed smooth and orientable real chain γ of ℂℙ_{2} is, with adequate orientation, the boundary of some holomorphic chain of ℂℙ_{2}\γ. These conditions have been qualified as mysterious in [19] because the functions satisfying these relations are produced "deus ex machina". The following criterion, which completes for a closed connected curve γ the one of [12], is obtained in [19] : Suppose that the second coordinate f_{2} of ℂ^{2} does not vanish on γ, then there exists in ℂℙ_{2}\γ a connected complex curve X with boundary ±γ in the sense of currents if and only if there exist p ∈ ℕ and A_{d} in the space O (d) of holomorphic homogeneous polynomials of order d, 1 ⩽ d ⩽ p, such that for ξ_{0} in some neighborhood of 0, C_{m} (ξ_{0}) = \frac{1}{2πi} ∫_{γ} \frac{f_{1}^{m}}{f_{2}+ξ_{0}} df_{2} satisfies

C_{k} = Q_{k,p} (C_{1}, ..., C_{p}) mod O (k) , k > p
C_{d} (ξ_{0}) = ∑_{k>d} \frac{(−ξ_{0})^{k}}{2πi} ∫_{γ} \frac{f_{1}^{d}}{f_{2}^{k+1}} df_{2} + A_{d} (ξ_{0}) , 1 ⩽ d ⩽ p

where Q_{k,p} are universal homogeneous polynomials.

In section 6.1, theorem 3a is develop for p > 0 into theorem 4 below which gives a more effective criterion for the Plateau problem in ℂℙ_{2} and also for the IDN-problem. This new criterion follows from considerations on sums of shock wave functions modulo affine functions in ξ_{0}. Even if decompositions in sum of shock wave functions are studied for ξ_{0}-affine functions, theorem 4 does not consider the case where G is of that type since it corresponds to a plain case of (1.4).

Notations for theorem 4 : if H and u are holomorphic functions on a simply connected domain D, we set D_{H} = \frac{∂}{∂ξ_{1}} − \frac{∂H}{∂ξ_{0}} and denote by L_{H}u the unique function v ∈ O (D) such that ∂v/∂ξ_{0} = D_{H}u and v (0, .) = 0 ; π_{1} is the projection (ξ_{0}, ξ_{1}) ↦ ξ_{1}.

Theorem 4. Let f be defined by (1.2) and consider the function G defined by (1.3). We assume that γ is connected and that f_{2} does not vanish on γ so that G, which is assumed to be not affine in ξ_{0}, is defined in a simply connected neighborhood D of 0 in ℂ^{2}.

A/ If (\overline{X} , γ) is a Riemann surface with almost smooth boundary where f extends meromorphically, then the following assertions hold for G

(1) There is p ∈ ℕ^{∗} and holomorphic functions a, b, λ_{1}, ..., λ_{p−1} on Δ = π_{1} (D) such that the integro-differential equation

−D_{G+L}L_{G+L}^{p−1} (G + L) + ∑_{1⩽j⩽p−1} D_{G+L}L_{G+L}^{p−1−j} λ̃_{j} = 0,

is valid with L = ξ_{0} ⊗ a + 1 ⊗ b and λ̃_{j} = 1 ⊗ λ_{j}, 1 ⩽ j ⩽ p.

(2) For s_{k} = −L_{G+L}^{k−1}G + L_{G+L}^{k−2}λ̃_{1} + ⋯ + L_{G+L}^{0}λ̃_{k−1}, 1 ⩽ k ⩽ p, the discriminant of T_{ξ_{0},ξ_{1}} = X^{p} + Σ_{1⩽k⩽p}s_{k} (ξ_{0}, ξ_{1}) X^{p−k} does not vanish

identically in D.

(3) G = −s_{1} − L (4) There is q ∈ ℕ, α, β ∈ ℂ_{q} [ξ_{1}] such that α (0) = 0, deg β < q and

a = \frac{α′}{1−α}, b = \frac{β}{1−α} Moreover, if p is the least integer such that (1-2-3) assertions holds, (γ, f)

uniquely determines (a, b, λ_{1}, ..., λ_{p−1}).

B/ Assume (1-2-3) holds for some p ∈ ℕ^{∗}. Then, there exists an open Riemann surface X such that \overline{X} = X ∪ γ is a manifold with almost smooth boundary where f extends meromorphically. Moreover, (4) holds. Remarks. 1. Non unicity of (a, b, λ_{1}, ..., λ_{p−1}) solving (1-2-3) means that X exists but p is not minimal.

2. It is possible that regardless its orientation, γ is the almost smooth boundary of an open Riemann surface X where f extends meromorphically. It is the case when γ cuts a compact Riemann surface Z into two smooth domains and f is the restriction to γ of an analytic map from Z to ℂℙ_{2}.

2. Intrinsic EIT on Riemann surfaces

The Inverse Dirichlet-Neuman problem, which goes back to Calderon [9] and which is called now Electrical-Impedance-Tomography problem, can be sketch like this : suppose that a bounded domain X in ℝ^{2} or ℝ^{3} is an ohmic conductor which means that the density of current j it may have is proportional (in isotropic cases) to the electrical field e = ∇U where U is an electrical potential. The scalar function σ such that j = σe is then called the conductivity of X ; ρ = 1/σ is the resistivity. When there is no time dependence and no source or sink of current, the equation div j = 0 holds and Calderon’s problem is then to recover σ on the whole of X from the operator C^{∞} (γ) ∋ u ↦ (σ∇ũ)_{γ}, ũ being the unique solution of div (σ∇ũ) = 0 with boundary value u.

In what follows, linking the Calderon problem to the Belishev problem mentioned in the introduction, we formulate the EIT-problem for a more general setting than the case of domains in ℝ^{n}. The second part of this section, despite the fact it is also quite elementary, seems to be new and underline how complex structure is involved in the dimension two case.

General dimension. Assume that X, an open oriented bordered manifold of dimension n with boundary γ, is given with a volume form μ and a conductivity σ modelled as a tensor from T^{∗}X to Λ^{n−1}T^{∗}X (see [27]). The gradient associated to σ relatively to μ is the differential operator which to any f ∈ C^{1} (X) associates the tangent vector field ∇_{μ,σ}f characterized by

(∇_{μ,σ}f) ⌟ μ = σ (df)

where ⌟ is the interior product. When U ∈ C^{1} (X) is some given potential, the density of physical current J is by definition^{(}^{10}^{)}

J = ∇_{μ,σ}U.

If X has no source or sink of currents and if U has no time dependence, the flux of current through the boundary of any domain is zero. Using Stokes formula, this can be modeled by the simplified Maxwell equation

(2.1)  0 = div_{μ} J = div_{μ} (∇_{μ,σ}U)

10That ∇_{μ,σ}U truly models the density of current is the assumption of Ohm’s law.

where div_{μ} is^{(}^{11}^{)} the divergence with respect to the volume form μ. Going back to the definition of gradient and divergence, we see that (2.1) is equivalent to the intrinsic equation formulated in [27] for domains in ℝ^{n}

(2.2)  d σ (dU) = 0

Since μ is no longer involved, the usual DN-operator has to be replaced by the operator Θ which to u ∈ C^{1} (γ) associates σ (du)_{γ} which is a section of Λ^{n−1}T_{γ}^{∗}X. The Electrical Impedance Tomography problem, is then to reconstruct (X , σ) from its DN-map Θ. Of course, the two other aspects of this problem, unicity and characterization, also has to be studied.

The problem in such a generality is still widely open ; almost all publications are about domains in ℝ^{3}. In such a case, (2.1) is generally written in euclidean global coordinates. However, when X is a manifold, (2.2) yields the same equation in any chart (W, x) ; setting σdx_{j} = Σ_{1⩽k⩽n}σ_{kj} (−1)^{k} dxk̂

with dxk̂ = ∧_{j≠k}dx_{j}, (2.2) becomes

(2.3)  ∑_{1⩽k⩽n} ∑_{1⩽j⩽n} \frac{∂}{∂x_{k}} (σ_{k,j} \frac{∂U}{∂x_{j}} ) = 0

When the conductivity σ is symmetric (σ (a) ∧ b ≡ σ (b) ∧ a) and invertible tensor, it is possible to design a natural metric g_{μ,ρ} associated to the resistivity map ρ = σ^{−1} by the well defined quotient of n-forms :

(2.4)  g_{μ,σ^{−1}} (t) = \frac{σ^{−1} (t ⌟ μ) ∧ (t ⌟ μ)}{μ}, t ∈ T X.

If (W, x) is any coordinates chart for X, a direct calculus in x-coordinates shows that for t = Σt_{k}∂/∂x_{k} (2.4) becomes

(2.5)  g_{μ,ρ} (t) = ∑_{k,ℓ} t_{ℓ} t_{k} λρ_{k,ℓ}

where (ρ_{k,ℓ}) is the matrix of the resistivity ρ = σ^{−1} when at any given point z the chosen basis for Λ^{n−1}T_{z}^{∗}X and T_{z}^{∗}X are ( (−1)^{k} dxk̂) and (dx_{k}) respectively. When (σ_{j,k}) is positive definite, g_{μ,ρ} is a metric on X.

When n ⩾ 3, there is a specially adequate choice of metric and volume.

Proposition 5. Assume n ⩾ 3. Then one can correctly design a global volume form μ by letting it be defined by μ = [det (ρ_{k,ℓ})]^{\frac{−1}{n−2}} dx_{1} ∧ ⋯ ∧ dx_{n} in any coordinates chart (W, x) for X. For this specific volume form, σ is the Hodge star operator of g_{μ,ρ} and μ is the riemannian volume form of g_{μ,ρ}.

This statement, already pointed out by Bossavit and Lee-Uhlmann (see [3] and [21]) for domains in affine spaces, follows from calculus in coordinates.

The interest of proposition 5 is to state the strict equivalence between the IDN-problem for riemannian manifolds and the EIT-problem when n ⩾ 3. When dim X ⩾ 3 and \overline{X} is a riemannian real analytic manifold with boundary, Lassas and Uhlmann have proved in [21] that the DN-operator uniquely determine X and its metric.

11If t is a differentiable vector field, div_{μ} t is defined by d (t ⌟ μ) = (div_{μ} t) μ.

The two dimensional case. We now assume n = 2 and σ = ρ^{−1} is symmetric and positive so that (X , g_{μ,ρ}) becomes a riemannian manifold whose volume form is thereafter denoted by V_{μ,ρ}. Let us emphasize the complex structure associated to the conformal class of (X , g_{μ,ρ}) by choosing isothermal coordinates charts, that is holomorphic charts (see e.g. [28]). In such a chart (W, z),

g_{μ,ρ} = κ_{μ,ρ} (dx ⊗ dx + dy ⊗ dy) = Re (κ_{μ,ρ}dz ⊗ d\overline{z})

where x = Re z, y = Im z and κ_{μ,ρ} ∈ C^{1} (W, ℝ_{+}^{∗}). Hence, in these coordinates, (σ_{k,ℓ}) = s diag (1, 1) with κ_{μ,ρ} = λ/s and λ ∈ C^{1} (W, ℝ_{+}^{∗}) is defined by μ = λ dx ∧ dy = λ \frac{i}{2}dz ∧ d\overline{z}. Note that s is a global positive function on X since it is the well defined quotient of volume forms :

s = μ/V_{μ,ρ}

Note also that s does not depend on μ and that (2.2) finally evolutes into

(2.6)  d (sd^{c}U) = 0

where d^{c} = i ( \overline{∂} − ∂ ), \overline{∂} and ∂ being the usual global differential operators associated to the complex structure of the conformal class of (X , g_{μ,ρ}). Hence, we have proved the following which generalizes a result written by Sylvester [27] for domains in ℝ^{2}.

Proposition 6. Let X be a real two dimensional manifold equipped with a symmetric and positive tensor σ : T^{∗}X → T^{∗}X. Then, there is a complex structure on X and s ∈ C^{1} (X , ℝ_{+}^{∗}), called scalar conductivity, such that (2.2) is equivalent to (2.6).

The beginning of this paper has shown that the data ∂U/∂ν is equivalent to the data (∂U)_{γ} which don’t involve any metric. Since the knowledge of (∂U)_{γ} is equivalent to the knowledge of (sd^{c}U)_{γ}, we consider (sd^{c}U)_{γ} as the DN-datum. We can now state an intrinsic IDN-problem for two dimensional ohmic conductors ; for the sake of simplicity, we limit ourselves to manifolds with boundary and smooth datas.

A two dimensional ohmic conductor is a couple (X , ρ) where X is an open oriented bordered two dimensional real surface (with boundary γ),the conductivity σ = ρ^{−1} is a positive definite tensor from T^{∗}X to T^{∗}X and X is equipped with the complex structure associated to the riemannian metric g_{μ,ρ} defined by (2.4) where μ is any volume form of X. In this setting, the scalar conductivity is the function s = μ/V_{μ,ρ} where V_{μ,ρ} is the volume associated to g_{μ,ρ}. The DN-operator is the operator θ_{X ,ρ} defined by

θ_{X ,ρ} : C^{1} (γ) ∋ u ↦ (sd^{c}ũ)_{γ} ∈ T_{γ}^{∗}X

where ũ is the unique solution of the following Dirichlet problem

(2.7)  U |_{γ} = u & d (s d^{c}U) = 0.

The IDN-problem associated to this setting is threefold : Unicity. Assume that two dimensional ohmic conductors (X , ρ) and (X′, ρ′) share the same boundary γ and the same DN-operator θ. Is it true that there is a diffeomorphism φ : \overline{X} → \overline{X′} between manifolds with boundaries such that φ : X → X′ is analytic and s = s′ ∘ φ where s and s′ are scalar conductivities of X and X′ ? Reconstruction. Assume that (X , ρ) is a two dimensional ohmic conductor. How from its DN-operator one can reconstruct a two dimensional ohmic conductor (X′, ρ′) which is isomorphic (in the above sense) to (X , ρ). Characterization. Let γ be a smooth abstract real curve, L a complex line bundle along γ and θ an operator from C^{1} (γ) to the space of smooth sections of L. Find a non trivial necessary and sufficient condition on (γ, L, θ) which ensures that there exists a two dimensional ohmic conductor (X , ρ) such that L = Λ^{1,0}T_{γ}^{∗}X and θ = θ_{X ,ρ}.

All these problems are open. In the particular case of constant scalar conductivity σ, the Dirichlet problem (2.7) becomes

U |_{γ} = u & dd^{c}U = 0

where dd^{c} = i∂\overline{∂} is the usual Laplacian. Hence, our article gives with theorems 1 to 3c a rather complete answer to the EIT-problem with constant scalar conductivity.

Concerning the main results given in the literature about unicity, reconstruction and stability for the important case where X is a domain in ℝ^{2} but the scalar conductivity is not constant, see [8][22] and references therein. Note that the exact method of reconstruction for this case goes back to [24].

3. Unicity under existence assumption

The notations and hypothesis are taken from theorem and section 1 ; we equip X with a hermitian metric^{(}^{12}^{)} g. Hence, there is a compact subset σ of γ such that h^{1} (σ) = 0 and (\overline{X} , γ) is a manifold with boundary near each point of γ\σ.

When u ∈ C^{∞} (γ), prop. 12 implies that u has a continuous harmonic extension ũ with finite Dirichlet integral on X. An elementary calculus gives then that for a fixed continuous defining function ρ of γ, smooth on \overline{X} \σ, the operator L defined by (1.1) determines for all u ∈ C^{∞} (γ) the trace on γ of the holomorphic (1, 0)-form ∂ũ : ∂ũ = (Lu) |∂ρ|_{g}^{−1} ∂ρ on γ\σ. With (1.2), this implies that f is the restriction to γ of a function F = (F_{1}, F_{2}) meromorphic on X, smooth on γ\σ. Since (γ, u, θu) is assumed to be a restricted DN-datum for X, F is continuous in a neighborhood of γ in \overline{X}.

The proof of theorem 1 relies on the following lemmas which enable to see \overline{X} as a normalization of \overline{Y}.

Lemma 7. Set δ = f (γ). Then Y = F (X) \δ is a complex curve of ℂℙ_{2}\δ without compact component, which has finite mass and satisfies d [Y] = [δ]. Moreover, each regular point of \overline{X} has in \overline{X} a neighborhood V such that F : V → F (V) is diffeomorphism between manifolds with smooth boundary.

Proof. Since F is continuous in a neighborhood of γ in \overline{X}, Y is a closed set of ℂℙ_{2}\δ. As Y is also locally the image of a Riemann surface by an analytic map, Y is a complex curve of ℂℙ_{2}\δ. Since F_{∗} [X] is a locally flat current, the Federer support theorem (see [17, p. 316][14, 4.1.15 & 4.1.20]) produces a locally integrable function λ on Y such that F_{∗} [X] = λ [Y] on the regular part Y_{reg} of Y ; since d^{2} = 0, λ is locally constant. Since f embeds γ into ℂ^{2}, each point x in γ which is a regular boundary point of \overline{X} has a neighborhood V of x in \overline{X} such that F : V_{x} → F (V_{x}) is a diffeomorphism between classical manifolds with boundary. Let F_{W} be the restriction of F to the Riemann surfaces W = ∪V_{x} and W′ = F (W). The degree of F_{W} is at most 1 otherwise almost all points of W′ would have at least two different preimages which would imply that df is zero at almost all points of γ. So this degree is 1 and F_{∗} [X] = [Y] on W′. Hence, λ = 1 on each connected component of Y and d [Y] = [δ].

12Harmonicity does not depend of the chosen hermitian metric.

If Y contains a compact complex curve Z, F^{−1} (Z) is a complex curve in X without boundary and so is empty. The fact that Y has a finite mass follows from a theorem of Wirtinger (see [17, Lemma 1.5 p. 315]). □

As δ is smooth, the conclusion of lemma 7 implies, thanks to [18], that δ contains a compact set τ such that h^{1} (τ) = 0 and ( \overline{Y}, δ ) is manifold with boundary near points of δ\τ. The lemma below described how Y is near a point y of τ.

Lemma 8. Assume Y is a complex curve of ℂℙ_{2}\δ with finite mass satisfying d [Y] = ± [δ]. Let y be a point of σ and U a domain containing y. Then, among the components of Y ∩ U, C_{y,1}^{U}, ..., C_{y,m_{U}}^{U}, one, says C_{y,1}^{U}, satisfies d [C_{y,1}^{U}] = ± [δ] |_{U_{y}} whereas for j ⩾ 2, \overline{C_{y,j}^{U}} ∩ U is a complex curve of U.

Proof. [18, th. 4.7] implies that for each j there is n_{j} ∈ ℤ such that d [C_{y,j}^{U}] =

n_{j} [δ] on U. As d [Y] = ± [δ], Σd [C_{y,j}^{U}] = ±1 and at least one C_{y,j}^{U}, says C_{y,1}^{U},

is such that n_{j} ≠ 0. Because h^{1} (σ) = 0, δ ∩ U contains a point q not in σ. Then, if V is a sufficiently small ball centered at q, Y ∩ V is submanifold of V with boundary δ ∩ V and Y ∩ V has only one connected component which can be nothing else than C_{y,1}^{U} ∩V. Hence n_{1} = ±1. Since two different bordered Riemann surfaces of some open set of ℂℙ_{2} meet at most in a set of zero one dimensional Hausdorff measure, this implies that n_{j} = 0 for j ≠ 1. Thus, if j ⩾ 2, d [C_{y,j}^{U}] = 0 and with [17, th. 2.1 p. 337] we conclude that

\overline{C_{y,j}^{U}} ∩ U is a complex curve of U_{y}. □

If y ∈ δ, we denote by m_{y} the limit of m_{U} (see lemma 8) when the diameter of U goes to 0, U neighborhood of y ; if m_{y} ⩾ 2, then y ∈ τ. A point y of δ is called a strong singularity of \overline{Y} if y is not a regular point of C_{y,1}^{U} and a weak singularity of \overline{Y} if y is regular point of C_{y,1}^{U} but m_{y} ⩾ 2.

We denote by τ_{1} (resp. τ_{2}) the sets of points where \overline{Y} has weak (resp. strong) singularity. Then τ = τ_{1} ∪ τ_{2} and τ_{1} ∩ τ_{2} = ∅. Note that both τ_{1} and τ_{2} may contains points y where m_{y} ⩾ 2.

We denote by \overline{Y}_{sing} = Y_{sing} ∪ τ the singular locus of \overline{Y}, that is the set of points of \overline{Y} where \overline{Y} is not a smooth manifold with boundary and we set B = f (σ) ∪ \overline{Y}_{sing}, A = F^{−1} (B) and X_{∘} = X \F^{−1} (δ) = X \F^{−1} (τ_{2}).

Lemma 9. The map F : \overline{X} → \overline{Y} is a normalization in the following sense : F : X_{∘} → Y is a (usual) normalization and F : \overline{X} \A → \overline{Y}\B is a diffeomorphism between manifolds with boundary.

Proof. Since X_{∘} = X \F^{−1} (δ), the properness of F |_{X_{∘}} and the finiteness of its fibers are elementary. For each connected component C of X_{∘}\A, the degree m_{C} of F : C → F (C) as a Riemann surfaces morphism is finite and F_{∗} [C] = δ_{C} [F (C)]. Reasoning as in lemma 7’s proof, we get m_{C} = 1. As Y_{sing} contains all the points of Y which has more than one preimage by F, F : C → F (C) is thus an isomorphism. Let C′ be another connected component of X_{∘}\A and assume that F (C) and F (C′) meet at q. Since q ∉ B, the germs of F (C) and F (C′) at q are equal. This leads to F (C) = F (C′) which yields the contradiction d [Y] = 2 [δ] near regular boundary points of δ in bF (C). Hence, F : X_{∘}\A → Y\B = Y_{reg} is an isomorphism of complex manifolds. As X_{∘} ∩ A = F^{−1} (Y_{sing}) has empty interior, F : X_{∘} → Y is a usual normalization.

Set X̃ = \overline{X} \A, Ỹ = \overline{Y}\B, τ̃ = τ ∪f (σ) and δ̃ = δ\τ̃ ; by definition of B, Ỹ is a manifold with smooth boundary δ̃ = δ\τ̃ and X̃ has smooth boundary γ\σ̃ where σ̃ = f^{−1} (τ) = σ ∪ f^{−1} (τ). The map F : X̃ → Ỹ is onto by construction. It is injective because the maps F : X_{∘} → Y and f : γ → δ are so and because if x_{1} ∈ X and x_{2} ∈ γ have the same image y by F, then m_{y} ⩾ 2, y ∈ τ and x_{1}, x_{2} ∈ A. Since Ỹ\τ̃ = Y_{reg} and F : X_{∘}\A → Y_{reg} is a diffeomorphism, the fact that F : X̃ → Ỹ is a diffeomorphism between manifolds with boundary has only to be check locally near boundary points. If x ∈ γ\σ̃, then y = f (x) ∉ τ and the last conclusion of lemma 7 implies that there are open neighborhoods V and W of x and y in \overline{X} and \overline{Y} such that F : V → W is diffeomorphism between manifolds with boundary. □

3.1. Proof of theorem 1. Let L′ be the operator defined by (1.1) when N is changed for N′, let us denote F′ the meromorphic extension of f to X′ and let Y′ = F′ (X′) \δ where δ = f (γ). By lemma 7, the sets Y′ and Y are two complex curves of ℂℙ_{2}\δ which has no compact component and both are bordered by [δ] in the sense of currents. Hence they are identical by a consequence of a Harvey-Shiffman theorem (see [12, prop. 1.4.1]).

Taking in account lemma 9 and the fact that B ∩ Y = Y_{sing} = B′ ∩ Y, this implies that Φ = F^{−1} ∘ F′ is an analytic isomorphism between X′\F^{′−1} (Y_{sing}) and X′\F^{−1} (Y_{sing}). Using the properness of F : X_{∘} → Y and F′ : X′_{∘} → Y, we conclude that Φ extends holomorphically to X′. Likewise, Ψ = F^{′−1} ∘ F extends holomorphically to X. As Φ(Ψ(x′)) = x′ and Ψ(Φ(x)) = x for almost all x′ ∈ X′ and x ∈ X, the extension of Φ is an isomorphism from X to X′.

As F and F′ extend f to X and X′, Φ extend continuously to γ by the identity map on γ . Set σ = \overline{X}_{sing} and σ′ = \overline{X′}_{sing} and let x′ be in γ\ (σ ∪ σ′). Then if y = f (x) ∉ τ = δ∩\overline{Y}_{sing}, Φ is a diffeomorphism between neighborhoods of x in \overline{X} and \overline{X′} because F (resp. F′) is a diffeomorphism between manifold with boundary from a neighborhood of x in \overline{X} (resp. in \overline{X′}) to a neighborhood of y in \overline{Y}. If y ∈ τ, then the last conclusion of lemma 7 implies that y ∈ τ_{1} so that there is a open neighborhood U of y, a component C_{y,1}^{U} of Y ∩ U and open neighborhoods V and V′ of x in \overline{X} and \overline{X′} such that that F : V → C_{y,1}^{U} and F′ : V′ → C_{y,1}^{U} are diffeomorphism between manifolds with smooth boundary. Hence, Φ : V → V′ is diffeomorphism between manifolds with smooth boundary. Finally, Φ realizes a diffeomorphism between manifolds with smooth boundary from X′\ (σ ∪ σ′) to X \ (σ ∪ σ′) and the proof is complete. _{■}

The proof contains the following variation of theorem 1.

Theorem 10. Assume that X and X′ are open Riemann surfaces with almost smooth boundary γ such that the map f defined by (1.2) is an embedding of γ into ℂℙ_{2} and has a meromorphic extension F to X and F′ to X′ which are continuous near γ. Then F (X) \f (γ) = F′ (X′) /f (γ) =^{def} Y is a complex curve of ℂℙ_{2}\δ without compact component, which has finite mass and satisfies d [Y] = [δ]. Moreover, \overline{X} and \overline{X′} are normalizations of \overline{Y} in the sense of lemma 9.

Thus, Riemann surfaces constructed in the converse part of theorems 3a, 3b and 3c are the only possible candidates for a solution to the IDN-problem.

4. Existence and reconstruction, proof of theorem 2

We first prove that the Stokes formula holds in almost smoothly bordered manifolds; one can see also [14].

Lemma 11. Let ( \overline{X} , γ ) be a Riemann surface with almost smooth boundary. Then for any 1-form φ which is continuous on \overline{X} such that dφ exists as an integrable differential on X, we have

(4.1)  ∫_{X} dφ = ∫_{γ} φ

Proof. Set σ = \overline{X}_{sing}. Since h^{2} ( \overline{X} ) < ∞ and h^{1} (σ) = 0, there is an increasing sequence (X_{k}) of smooth open sets of X such that (h^{1} (bX_{k})) and (h^{2} ( \overline{X} \X_{k}) ) both have limit zero and \overline{X} \\overline{X_{k}} is contained in a 2^{−k}- neighborhood of σ. Let φ be as above. Since dφ is integrable and lim h^{2} (X_{k}) = 0, (∫_{X_{k}} dφ) has limit ∫_{X} dφ. As bX_{k} = (γ ∩ \overline{X_{k}})∪[(bX_{k}) \γ] and lim h^{1} (bX_{k}) =

0, (∫_{(bX_{k})\γ} φ) converges to 0. Hence, lim ∫_{γ∩\overline{X_{k}}} φ = ∫_{γ} φ and the classical Stokes formula for φ and X_{k} yields (4.1). □

We prove now a variation of the Riemann’s existence theorem.

Proposition 12. Let ( \overline{X} , γ ) be a Riemann surface with almost smooth boundary and u a real valued lipschitzian function on γ. Then u has a unique continuous harmonic extension ũ of u to X and ũ has finite Dirichlet integral ∫ i ∂ũ ∧ \overline{∂}ũ. Moreover, N_{X} u defined as ∂ũ/∂ν on γ\\overline{X}_{sing} admits an extension on γ as a current of order 1 on γ.

Proof. Following the lines of Riemann’s method for harmonic extension of smooth functions, we first construct an adequate space W^{1} (X).

Since ( \overline{X} , γ ) is at least a topological bordered manifold, for every fixed point x in γ we can choose in X an open set Δ_{x} whose closure in \overline{X} is a neighborhood of x and which is mapped by a complex coordinate φ_{x} into the closure of the unit disk 𝔻 of ℂ, φ_{x} being a homeomorphism from \overline{Δ_{x}} to \overline{𝔻}. Note that if x′ ∈ γ ∩ \overline{Δ_{x}} is a regular point of \overline{X}, φ_{x} has to be diffeomorphism between manifolds with boundary from a neighborhood of x′ to a neighborhood of φ_{x} (x′) in \overline{𝔻}. If x ∈ X, we choose a conformal open disk φ_{x} : Δ_{x} → 𝔻 of X centered at x. With the help of a continuous partition of unity, we can now construct a continuous hermitian metric h on X by gluing together the local metrics (φ_{x})_{∗} dz ∧ d\overline{z} where z is the standard coordinate of ℂ. We then denote by W^{1} (X) the Sobolev space of functions in L^{2} (X , h) with finite Dirichlet integral.

By construction, any function A in W^{1} (X) is such that for each x ∈ γ, B_{x} = (φ_{x})_{∗} A |_{Δ_{x}} is square integrable for the standard metric of 𝔻. Since the values of Dirichlet integrals are conformal invariants, it follows that B_{x} is in the standard Sobolev space W^{1} (𝔻) and hence admits a boundary value b_{x} on 𝕋 = b𝔻 which is in W^{1/2} (𝕋). As b_{x} is punctually defined almost everywhere, a_{x} = b_{x} ∘ φ is defined almost everywhere in γ ∩ \overline{Δ_{x}}. The constructions made for each x ∈ γ glue together to form a function defined almost everywhere in γ which we call the boundary value of A.

We consider now the subset F of W^{1} (X) with boundary value u. It is closed and non empty since by a result of McShane [23], u admits a Lipschitz extension to X. It follows now from classical arguments that the Dirichlet integral can be minimized in F at some function ũ which has to be harmonic in X. It remains only to show that ũ is continuous on \overline{X}. If x ∈ γ, what precedes implies that v_{x} = (u_{x} ∘ φ_{x}^{−1}) |_{𝕋} is in W^{1/2} (𝕋), continuous near φ_{x} (x) and is the boundary value of ṽ_{x} = ũ ∘ φ_{x}^{−1}. Hence, classical Poisson formula for the disc implies that near φ_{x} (x) in \overline{𝔻}, ṽ_{x} is continuous up to 𝕋 with restriction v_{x} on 𝕋. Since φ_{x} is an homeomorphism, we get that ũ is continuous at x with value u (x).

Let θu be the form defined by 1.1. The Stokes formula (4.1) implies that if φ ∈ C^{1} (γ) and Φ is a Lipschitz extension of φ on X,

∫_{γ} φθu = − ∫_{X} ∂ũ ∧ \overline{∂}Φ.

As the last integral is independent of the Lipschitz extension of φ, this means that θu and hence N_{X} u, are well defined currents of order 1. □

Assume now that the hypothesis of theorem 2 are true. Lemma 7 points out that F projects γ on a smooth curve δ of ℂ^{2} which bounds in the sense of currents a complex curve Y of ℂℙ_{2}\δ which has finite mass and no compact component and theorem 10 implies that for some subset X_{∘} of X with discrete complement in X, F : X_{∘} → Y is usual normalization. Hence, if B = Y_{sing} and A = F^{−1} (B), F : X \A → Y\B is one to one. This is part 1 of theorem 2.

Before proving the second claim of theorem 2, we recall that ℂℙ_{2} is equipped with homogenous coordinates w and ℂ^{2} identified with {w_{0} ≠ 0} has affine coordinates z_{1} = w_{1}/w_{0} and z_{2} = w_{2}/w_{0}. Set Δ_{∞} = {w_{0} = 0}, Y_{∞} = Y ∩ Δ_{∞} and if ξ ∈ ℂ, we set Δ_{ξ} = {w_{2} = ξw_{0}} and Y_{ξ} = Y ∩ Δ_{ξ}. Set

Ω_{ξ}^{m} = \frac{z_{1}^{m}}{z_{2} − ξ} dz_{2} = \frac{w_{1}^{m}}{w_{0}^{m}} \frac{dw_{2}}{w_{2} − ξw_{0}} − \frac{w_{1}^{m}}{w_{0}^{m+1}} \frac{w_{2}dw_{0}}{w_{2} − ξw_{0}} .

Applying the Stokes formula either for Y or X, it turns out that \frac{1}{2πi} ∫_{γ} \frac{f_{1}^{m}}{f_{2}−ξ} df_{2} equals S_{m} (ξ) + P_{m} (ξ) where

S_{m} (ξ) = ∑_{z∈Y_{ξ}} Res (η^{∗}Ω_{ξ}^{m}, z) ; P_{m} (ξ) = ∑_{z∈Y_{∞}} Res (η^{∗}Ω_{ξ}^{m}, z)

and η : Y → ℂℙ_{2} is the canonical injection.

For almost all ξ_{∗} in ℂ, Y meets transversely Δ_{ξ_{∗}} only in ℂ^{2} ∩ Y_{reg} ; for such a fixed ξ_{∗}, set p = Card Y_{ξ_{∗}} and Y_{ξ_{∗}} = {z_{1∗}, ..., z_{p∗}}. For ξ in a sufficiently small connected neighborhood W_{ξ_{∗}} of ξ_{∗}, Y_{ξ} lies then in ℂ^{2}∩Y_{reg} and can be written {z_{1} (ξ) , ..., z_{p} (ξ)} with z_{j} (ξ) = (h_{j} (ξ) , ξ) where h_{j} is holomorphic in W_{ξ_{∗}} and has value z_{j∗} at ξ_{∗}, 1 ⩽ j ⩽ p. Direct calculation shows (see [12]) that the poles of Ω_{ξ}^{m} in ℂ^{2} are z_{1} (ξ) , ..., z_{p} (ξ) with residue h_{1} (ξ)^{m} , ..., h_{p} (ξ)^{m}. Hence, S_{m} = S_{h,m} in V.

Reasoning as in lemma 13 in next section, we can assume without loss of generality that Y meets transversely Δ_{∞} and that Y_{∞} ⊂ Y_{reg}. In this situation, a direct calculus (see [12]) gives that at y ∈ Y_{∞}, Ω_{ξ}^{m} has a pole of order m + 1 with a residue which is a polynomial in ξ of degree at most m. Hence, P_{m} is is a polynomial in ξ of degree at most m ; formula (E_{m,ξ}) is proved.

If A ⩾ B and ξ_{0}, ..., ξ_{A−1} are mutually distinct, the Vandermonde matrix (ξ_{ν}^{μ})_{0⩽ν,μ⩽B−1} is invertible and hence, the system (E_{m,ξ_{ν}})_{0⩽ν⩽B−1} enable to write the coefficients of P_{m} as a linear combination of the S_{h,m} (ξ_{ν}), 0 ⩽ ν ⩽ B − 1. Introducing this result in (E_{ξ}) = (E_{m,ξ_{ν}})\begin{matrix}0⩽m⩽B−1 \\ 0⩽ν⩽B−1 \end{matrix} we get a

linear system which, since AB− \frac{1}{2}B (B + 1) ⩾ pA when B ⩾ 2p+1, enable to compute for a generic ξ the unknowns S_{h,m} (ξ_{ν}) and, thanks to the Newton-Girard formulas, the elementary symmetric functions of h_{1} (ξ_{ν}) , ..., h_{p} (ξ_{ν}) ; finally we get the intersection points (h_{j} (ξ_{ν}) , ξ_{ν}) of Y with Δ_{ξ_{ν}}.

We prove the third assertion of theorem 2. Almost all ξ_{∗} in ℂ^{2} has a connected neighborhood W_{ξ_{∗}} such that there is a compact of ℂℙ_{2}\ (δ ∪ Y_{sing}) containing all Y_{ξ} when ξ ∈ W_{ξ_{∗}}. When ξ_{∗} is such and ξ ∈ W_{ξ_{∗}} the form Φ_{ξ}^{m,ℓ} = \frac{F_{1}^{m}}{F_{2}−ξ} ∂ũ_{ℓ} may have poles of order at most m at infinity i.e. in {w_{0} = 0} ∩ X and whiles its over poles lies in a compact of X. Since ũ_{ℓ} is the continuous harmonic extension of u_{ℓ} on X, ∫_{X} i ∂ũ_{ℓ} ∧ \overline{∂}ũ_{ℓ} < +∞ by proposition 12 and we can apply the Stokes formula (4.1) to it on X. This gives (T_{m,ξ}) after a residue calculus.

Remark. The \frac{1}{2}B (B + 1) coefficients of the polynomials P_{k} come from the residues of the intersection points of Y with {w_{0} = 0}. In the generic case, Y is given near theses points as the graph of holomorphic functions ψ_{1}, ..., ψ_{q} of the variable w_{0}/w_{2}, it appears that the coefficients of P_{k} are ruled by the derivatives of order at most k at 0 of the ψ_{ℓ}. The reconstruction of X is thus possible with a non linear system with only pA + q (B + 1) unknowns.

5. Proofs of characterizations theorem 3a, 3b and 3c

The proofs of theorems 3a, 3b and 3c follow a similar schema. The function f defined by (1.2) embeds γ into a smooth real curve δ = f (γ) of ℂ^{2}. The necessary conditions for the existence of a solution to the IDN-problem for γ are drawn from the fact this existence implies that δ bounds a "concrete" Riemann surface in ℂℙ_{2} or ℂ^{2}. The sufficient part of theorem 3c reconstructs the concrete but singular solution to the IDN-problem ; a normalization gives then the sufficient part of 3b. The proof of theorem 3a follows a similar scheme.

5.1. Proof of A.theorem 3a. Assume that X is an open bordered riemannian surface of finite volume with restricted DN-datum (γ, u, θu). Then the functions F_{j} (j = 1, 2) which are the well defined quotients of forms (∂ũ_{j}) / (∂ũ_{0}) are meromorphic and letting F = (F_{1}, F_{2}), lemma 7 implies that Y = F (X) \δ, δ = f (γ), is a complex curve of finite volume, without compact component and bordered by [δ] in the sense of currents. Moreover, the function G has the expression

G (ξ_{0}, ξ_{1}) = \frac{1}{2πi} ∫_{δ} Ω_{ξ} , Ω_{ξ} = \frac{w_{1}}{w_{0}} \frac{dΛ_{ξ} (w)}{Λ_{ξ} (w)} − \frac{w_{1}}{w_{0}^{2}} dw_{0} ,

where δ = f (γ), (w_{0} : w_{1} : w_{2}) are homogenous coordinates for ℂℙ_{2} and Λ_{ξ} (w) = ξ_{0}w_{0} + ξ_{1}w_{1} + w_{2}.

For almost all ξ_{∗} = (ξ_{0∗}, ξ_{1∗}) and for all ξ in a sufficiently small connected neighborhood W_{ξ_{∗}} of ξ_{∗}, Y meets Δ_{ξ} = {Λ_{ξ} = 0} transversely, Y_{ξ} = Y ∩ Δ_{ξ} ⊂ ℂ^{2} ∩ Y_{reg} so that there exists p = Card Y_{ξ_{∗}} holomorphic functions H_{j} = (1 : h_{j} : h_{j,2}) : W_{ξ_{∗}} → ℂℙ_{2} such that Y_{ξ} = {H_{j} (ξ) , 1 ⩽ j ⩽ p} and (h_{j})_{1⩽j⩽p} are mutually distinct. Direct calculations^{(}^{13}^{)} shows that these functions satisfy the shock wave equation (1.5).

Let η : Y → ℂℙ_{2} the canonical injection. Since η^{∗}Ω_{ξ} may only have poles in Y_{ξ} ∪ Y_{∞}, the Stokes formula gives that near ξ_{∗}, G = H + L where

H (ξ) = ∑_{z∈Y_{ξ}} Res (η^{∗}Ω_{ξ}, z) , L (ξ) = ∑_{z∈Y_{∞}} Res (η^{∗}Ω_{ξ}, z) .

By construction, η^{∗}Ω_{ξ} has residue h_{j} (ξ) at z = H_{j} (ξ) ∈ Y_{ξ} and it remains only to know that L is affine in ξ_{0} to prove theorem 3a. The second part of the lemma below is needed in the proof of theorem 4.

Lemma 13. If W_{ξ_{∗}} is small enough, L = Σh_{j} − G is affine in ξ_{0}. In addition, there is an integer q such that L is the limit in O (W_{ξ}^{∗}) of a continuous one parameter family of ξ_{0}-affine functions which are sum of q mutually distinct shock wave functions.

Proof. With no loss of generality, we assume ξ_{∗} = 0 for the proof. For small complex parameters ε, we consider the homogeneous coordinates w^{ε} = (w_{0} + εw_{1} : w_{1} : w_{2} + εw_{1}). For ε in a sufficiently small neighborhood of 0 the intersection of Y with the zero set of Λ_{ξ} : w ↦ ξ_{0} + ξ_{1}w_{1}^{ε} + w_{2}^{ε} is still generic in the sense that it is transverse and lies in {w_{2}^{ε} ≠ 0} ∩ Y_{reg}. Hence,

setting Ω_{ξ}^{ε} = \frac{w_{1}}{w_{0}^{ε}} \frac{dΛ_{ξ}^{ε}(w)}{Λ_{ξ}^{ε}(w)} −_{\frac{w_{1}}{(w_{0}^{ε})^{2}}} dw_{0}^{ε}, the function G^{ε} : ξ ↦ \frac{1}{2πi} ∫_{δ} Ω_{ξ}^{ε} is, on W_{ξ_{∗}}

the sum of p mutually distinct shock wave functions h_{1}^{ε}, ..., h_{p}^{ε}. For generic ε, Y meets transversely Δ_{∞}^{ε} = {w_{0}^{ε} = 0} and Y_{∞}^{ε} = Y ∩ {w_{0} = 0} lies in Y_{reg} ∩ {w_{2}^{ε} ≠ 0}. Hence, [12, Lemme 2.3.1] implies that L^{ε} = Σh_{j}^{ε} − G^{ε} is affine in ξ_{0}. The dependence of G^{ε} is clearly holomorphic in ε. The same holds for each h_{j}^{ε} since what precedes have shown that h_{j}^{ε} (ξ) = Res (η^{∗}Ω_{ξ}^{ε}, H_{j} (ξ)) =

13This lemma which goes back to Darboux is proved in [12, lemma 2.4].

\frac{1}{2πi} ∫_{Y∩∂U_{j}} η^{∗}Ω_{ξ}^{ε} where U_{j} is any sufficiently small neighborhood of H_{j} (ξ) in ℂℙ_{2} whose boundary is smooth and transverse to Y. Hence L^{ε} is holomorphic in ε and has to be affine in ξ_{0} when ε = 0.

Let q be the number of points in Y_{∞} counted with their multiplicities ; when ξ is generic, q is either defined by

(5.1)  p − q = \frac{1}{2πi} ∫_{γ} \frac{d (ξ_{0} + ξ_{1}f_{1} + f_{2})}{ξ_{0} + ξ_{1}f_{1} + f_{2}}.

For sufficiently small generic ε, [12, Lemme 2.3.1] gives more precisely that

L^{ε} = Σ_{1⩽j⩽q}h_{j}^{ε,∞} with

h_{j}^{ε,∞} = − Res (η^{∗}Ω_{ξ}, z_{j}^{ε}) = \frac{−ξ_{0}ψ_{j}^{ε} (0) + ψ_{j}^{ε}′ (0)}{1 + ξ_{1}ψ_{j}^{ε} (0)}

where Y_{∞}^{ε} = {z_{1}^{ε}, ..., z_{q}^{ε}} and ψ_{j}^{ε} ∈ O (U^{ε}), U^{ε} open neighborhood of 0 in ℂ, enable to give in the affine coordinates ζ^{ε} = (w_{j}^{ε}/w_{2}^{ε})_{j=0,1} the set Y as a graph above U^{ε} : Y ∩ V_{j}^{ε} = {(ζ_{0}^{ε} : ψ_{j}^{ε} (ζ_{0}^{ε}) : 1) ; ζ_{0}^{ε} ∈ U^{ε}}. Each h_{j}^{ε,∞} is clearly a shock wave function, that is a solution to h_{ξ_{1}} = h_{ξ_{0}}h. □

Remark. When ε goes to a non generic value, the fact that L is a sum of q shock wave functions may not be preserved as section 6.1 shows.

5.2. Proof of B.theorem 3a. Assume that γ satisfies (1.4) in a connected neighborhood W_{ξ_{∗}} of one point (ξ_{0∗} : ξ_{1∗} : 1) of ℂℙ_{2}. If (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0, then γ satisfies the classical Wermer-Harvey-Lawson moment condition in ℂ_{ξ_{∗}}^{2} = ℂℙ_{2}\ {ξ_{0∗}w_{0} + ξ_{1∗}w_{1} + w_{2} = 0} (see [12, cor. 1.6.2]) and [29][18] implies that if δ is suitably oriented, the polynomial hull of δ in ℂ_{ξ_{∗}}^{2} is the unique complex curve Y of finite mass of ℂ_{ξ_{∗}}^{2}\δ such that d [Y] = [δ].

Assume now (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0. Then we can choose a minimal H = {h_{1}, ..., h_{p}} in the sense that no proper subset of H satisfy (1.4). Although it is not explicitly mentioned by their authors, the heart of the arguments of [12, th. II p. 390] is that ± [δ] = d [Y] where Y is the analytic extension Y in ℂℙ_{2}\δ of the union Γ of the graphs Γ_{j} of the functions

H_{j} : ξ ↦ (1 : h_{j} (ξ) : −ξ_{0} − ξ_{1}h_{j} (ξ)) , 1 ⩽ j ⩽ p.

This fact, not totally explicit in [20, p. 264], can be recovered a posteriori by a kind of trick which has been used Poly in [13] and is developed later in the proof of theorem 3c : for the curve γ̃ which is the union of γ with the boundaries of Γ_{j} negatively oriented, one goes back to the ℂ_{ξ_{∗}}^{2}-case where (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0. If d [Y] is − [δ] and not [δ], then the same arguments which have proved theorem 3a.A would give that the functions h_{j}, geometrically defined as the first coordinates of points of intersection of Y with generic lines Λ_{ξ}, should satisfy not only the shock wave equation h_{ξ_{1}} = h_{ξ_{0}}h but also the "negative" shock wave equation h_{ξ_{1}} = −h_{ξ_{0}}h. As this is impossible, d [Y] = [δ].

In both cases, we have found (up to a change of orientation if ∂^{2}G/∂ξ_{0}^{2} vanish on W_{ξ_{∗}}) a complex curve Y of finite mass of ℂℙ_{2}\δ such that d [Y] = [δ].

As δ is smooth, we know from [18] that there is in δ a compact set τ such that h^{1} (τ) = 0 and for which each point of y ∈ δ\τ has a neighborhood U_{y} where \overline{Y} ∩ U_{y} is a closed bordered submanifold of U_{y} with boundary δ ∩ U_{y}. Lemma 8 in section 3 describes how Y is near points of τ. Using the notations and definitions introduced after its proof, we let τ_{2} (resp. τ′) be the set of y in δ where \overline{Y} has a strong singularity (resp. m_{y} ⩾ 2) and define Ỹ as the abstract complex curve Y ∪ τ′.

Consider a normalization π : X → Ỹ ; lemma 8 implies that π is an open mapping. Let Z be the disjoint and abstract union X ∪ γ. If x ∈ γ, we define a neighborhood of x in Z as a subset of Z which contains a set of the kind π^{−1} (C_{y,1}^{U}) where y = f (x) and U is a neighborhood of y in ℂℙ_{2}. Then (Z, γ) is a compact metrizable topological manifold with boundary which has finite 2-dimensionnal Hausdorff measure and smooth boundary outside σ = f^{−1} (τ_{2}). Since f is an embedding, h^{1} (σ) = 0 and (Z, γ) is a manifold with almost smooth boundary. Moreover, it follows by construction that the meromorphic extension F : Z → \overline{Y} of f to X defined by F |_{X} = π is a normalization of \overline{Y} in the sense of lemma 9._{□}

Remark. When γ is real analytic, [18, th. II] implies that C_{y,1}^{U} is a manifold with boundary in the classical sense. So, in that case, (Z, γ) is a classical manifold with boundary.

The following proposition which clarifies some results of [12] justifies the fourth remark after theorem 3a.

Proposition 14. Assume δ is connected. Then if (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0, the

polynomial hull of γ in ℂ^{2} = ℂℙ_{2}\ {ξ_{0∗}w_{0} + ξ_{1∗}w_{1} + w_{2} = 0} has boundary ± [γ]. If (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0 and no proper subset of H = {h_{1}, ..., h_{p}} satisfies (1.4), then the analytic extension Y in ℂℙ_{2}\δ of the union Γ of the graphs Γ_{j} of the functions H_{j} : ξ ↦ (1 : h_{j} (ξ) : −ξ_{0} − ξ_{1}h_{j} (ξ)), 1 ⩽ j ⩽ p, is the complex curve which has minimal volume among complex curves Z such that d [Z] = [δ].

Proof. The preceding proof contains the above conclusion except the minimality of volume of Y when (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0 and H is minimal. In that

case, let Y′ be a complex curve of ℂℙ_{2}\δ with boundary [δ] and minimal volume. Then Y = Y′ ∪ Z where Z is a union of compact complex curves of ℂℙ_{2}. But the intersection of any compact curve with a line Λ_{ξ} is described, for ξ in a neighborhood of a generic ξ_{∗}, as a finite union of graphs of function (1 : g_{j} : g_{j,2}) whose second homogeneous coordinate satisfy the shock wave equation and such that Σg_{j} is affine in ξ_{0} (see [20, section 2]). Since H is minimal, no such g_{j} belongs to H and it appears that Γ has to be contained in Y′ Hence, Y ⊂ Y′ and, finally, Y = Y′. □

5.3. Proof of C.theorem 3a. Let X, Z, D and g be as in the statement. Let u ∈ C^{1} (γ) and for z ∈ D\γ set

Ω_{z} = u∂_{ζ}g_{z} + g_{z}\overline{θu}.

If Y is a smooth domain in Z, the Stokes formula (4.1) implies that value of 1_{Y} (z) û (z) + ∫_{Y} \overline{∂}û ∧ ∂_{ζ}g_{z} − ∫_{(∂Y)\γ} u∂_{ζ}g_{z} does not depend of the Lipschitz extension û of u to Z and equals ∫_{γ∩Y} u∂_{ζ}g_{z} when X_{sing} = ∅. Hence we can take it as a definition for ∫_{γ∩Y} u∂_{ζ}g_{z} in the general case. Let then F be the function defined for z ∈ D\γ by

F (z) = \frac{2}{i} ∫_{ζ∈γ} Ω_{z} (ζ) , Ω_{z} = u∂_{ζ}g_{z} + g_{z}\overline{θu},

where g_{z} = g (., z).

Since ũ, u and θu are continuous, the conclusion of part C follows from the lemma below which gives ũ = F |_{X} and θu = ∂ũ on γ\σ if F |_{D\\overline{X}} = 0.

Lemma 15. F_{+} = F |_{X} and F_{−} = F |_{D\\overline{X}} are real valued harmonic functions such that

(5.2)  u = F_{+} − F_{−} & \overline{θu} = \overline{∂}F_{+} − \overline{∂}F_{−} on γ\σ.

Proof. The harmonicity of F is a simple consequence of the properties of g. Fix now p in γ\σ and in neighborhood U of p in D, a holomorphic chart U → U centered at p ; a hat " " denotes thereafter the coordinate expression of a function, a form or a set. For z ∈ U\γ let us write F (z) = φ (z)+R_{1} (z) with φ (z) = −2i ∫_{γ∩U} Ω_{z} and R_{1} is smooth on U. If y and x are the coordinates of ζ ∈ γ ∩ U and z ∈ U\γ, ĝ(y, x) = g (ζ, z) can be written in the form ĝ(y, x) = \frac{1}{2π} ln |y − x| + h (y, x) where h is a smooth function on U × U, harmonic in each its variable. Hence

φ̂ (x) = \frac{1}{2πi} ∫_{γ̂∩U} \frac{û (y)}{y − x}dy + ∫_{γ̂∩U} \frac{ln |y − x|}{πi}\overline{θ̂u (y)} + R_{2} (x)

where R_{2} ∈ C^{0} (U). The second integral has no jump across γ̂ and from the classical Sohotsky-Plemelj formula, we know that the first integral has jump û across γ̂ in the sense of distribution and pointwise near each regular boundary point. Since

\overline{∂}φ̂ (x) = \frac{1}{2πi}d\overline{x} ∫_{γ̂∩U} \frac{1}{\overline{x} − \overline{y}} \overline{θ̂u (y)} + \overline{∂}R_{2} (x) ,

the jump of \overline{∂}φ̂ through γ̂ is likewise \overline{θ̂u}.

In order to check that F (z) ∈ ℝ when z ∈ D\γ, we let L_{X} be the DNoperator of X and we note that since ∂g_{z} = (L_{X} g_{z}) (ν^{∗} + iτ^{∗}) and θu = (Lu) (ν^{∗} + iτ^{∗}), − Im F (z) = ∫_{γ} (u τg_{z} + g_{z}τu) τ^{∗} = ∫_{γ} d (ug_{z} |_{γ}) = 0. □

5.4. Proof of A.theorem 3c. We assume that γ is in the sense of currents the boundary of a complex curve X of ℂℙ_{2}\γ for which (γ, u, θu) is a restricted DN-datum. Then as in the proof of theorem 3a.A, for almost all (ξ_{∗0} : ξ_{∗1} : 1) in ℂℙ_{2} there exists a neighborhood W_{ξ_{∗}} of ξ_{∗} = (ξ_{∗0}, ξ_{∗1}) such that for every ξ ∈ W_{ξ_{∗}}, X ∩ Δ_{ξ} lies in ℂ^{2} and equals Γ ∩ Δ_{ξ} where Γ is the union of the graphs Γ_{j} of H_{j} = (1 : h_{j} : h_{j,2}) : W_{ξ_{∗}} → ℂℙ_{2}, 1 ⩽ j ⩽ p where H_{j} is holomorphic in W_{ξ_{∗}}. Since we are concerned only by generic ξ_{∗}, we can suppose that Γ_{j} = {(φ_{j} (z_{2}) , z_{2}) ; z_{2} ∈ U_{j}} where U_{j} is a neighborhood of z_{j,2}^{∗} = h_{j,2} (ξ_{∗}) and φ_{j} ∈ O {U_{j}}. The decomposition sought for G̃ in (a1) can be then found in [20]. However, this residues calculus is needed in part B and we include it here

When ξ ∈ W_{ξ_{∗}}, G̃_{ℓ} (ξ_{0} : ξ_{1} : 1) is the sum of the residues of

Λ_{ℓ} = \frac{z_{ℓ}}{ξ_{0} + ξ_{1}z_{1} + z_{2}} Θ_{0}

(for convenience z_{ℓ} = 1 if ℓ = 0) in X. Set Θ_{0} = A_{j} (z_{2}) dz_{2} in each Γ_{j} and let us abbreviate H_{j} (ξ) in z_{j}. Then z_{j,2} is the only pole of Λ_{ℓ} in Γ_{j}. It is a simple one and the residue of Λ_{ℓ} at it is

(5.3)  g_{j,ℓ} = \frac{z_{j,ℓ} A_{j} (z_{j,2})}{ξ_{1}φ′_{j} (z_{j,2}) + 1} = z_{j,ℓ} g_{j,0}

where z_{j,ℓ} = 1 if ℓ = 0. As Γ_{j} is also parametrized by H_{j}, we can set

g_{j} = Σ_{0⩽ℓ⩽2}g_{j,ℓ} dη_{ℓ} on Γ_{j}

and get that g_{j,1}/g_{j,0} = z_{j,1} = h_{j} satisfy (1.5) ; g_{j,2}/g_{j,0} = z_{j,2} = h_{j,2} satisfy then h_{j} \frac{∂h_{j,2}}{∂ξ_{0}} = \frac{∂h_{j,2}}{∂ξ_{1}} because h_{j,2} = −ξ_{0} − ξ_{1}h_{j}. Note that the dependence in ξ of g can be made more clear if the identity h_{j} − φ_{j} (−ξ_{0} − ξ_{1}h_{j}) = 0 is used. Indeed, this relation implies

(1 + ξ_{1}φ′_{j}) ∂_{ξ_{0}}h_{j} + φ′_{j} = 0 & (1 + ξ_{1}φ′_{j}) ∂_{ξ_{1}}h_{j} + h_{j}φ′_{j} = 0

Hence φ′_{j} = − (∂_{ξ_{0}}h_{j}) / (1 + ξ_{1}∂_{ξ_{0}}h_{j}) and

\frac{1}{1 + ξ_{1}φ′_{j}} = \frac{∂_{ξ_{1}}h_{j}}{∂_{ξ_{0}}h_{j}} \frac{1 + ξ_{1}∂_{ξ_{0}}h_{j}}{h_{j}} = 1 + ξ_{1}∂_{ξ_{0}}h_{j} = ∂_{ξ_{0}}h_{j,2}

since h_{j} satisfies h_{j}∂_{ξ_{0}}h_{j} = ∂_{ξ_{1}}h_{j}. So we have now instead of (5.3)

(5.4)  g_{j,ℓ} = A_{j} (H_{j}) h_{j,ℓ} \frac{∂h_{j,2}}{∂ξ_{0}} , 1 ⩽ j ⩽ p, 0 ⩽ ℓ ⩽ 2.

where h_{j,0} = 1 for convenience.

From the definition we get that when expressed in the affine coordinates ξ, g_{j} is given by the integral formula

2πi g_{j} = (∫_{∂Γ_{j}} \frac{Θ_{0}}{ξ_{0} + ξ_{1}z_{1} + z_{2}} ) dξ_{0} + (∫_{∂Γ_{j}} \frac{z_{1}Θ_{0}}{ξ_{0} + ξ_{1}z_{1} + z_{2}} ) dξ_{1}

from which it is clear that g_{j} is closed.

To achieve the proof of part A, it is enough to remark that (a2) is a direct consequence of the fact Re Θ_{ℓ} = dũ_{ℓ} is exact.

5.5. Proof of B.theorem 3c. Assume that the hypothesis of (b1) is true and γ is connected.

Case G̃ |_{W_{η_{∗}}} ≠ 0. This mean H ≠ ∅ when H is minimal in the sense that no proper subset of H gives a decomposition of G̃ with the same properties. Let Γ be the union of the graphs Γ_{j} of the functions H_{j} = (1 : h_{j} : h_{j,2}), 1 ⩽ j ⩽ p, where h_{j,2} = −ξ_{0} − ξ_{1}h_{j}. If needed, we can choose another ξ_{∗} in order that Γ does not meet Λ_{ξ_{∗}} in {w_{0} = 0}. Then for any ξ in a neighborhood Ω of ξ_{∗}, the H_{j} (ξ) are mutually distinct and are the points of Γ ∩ L_{ξ}. Finally, we assume, which it is not a restriction, that Γ has a smooth oriented boundary ∂Γ.

Let γ̃ be the union of ∂Γ with opposite orientation and γ and let φ_{η} be the linear function z ↦ η_{0}+η_{1}z_{1}+η_{2}z_{2}. From the hypothesis we get directly

\frac{1}{2πi} ∫_{γ} φ_{η}^{−1}θ_{ℓ} = ∑_{1⩽j⩽p} g_{j,ℓ} = ∑_{1⩽j⩽p} g_{j,ℓ}h_{j,ℓ}

where h_{j,ℓ} = 1 if ℓ = 0. On the other hand, if we set

(5.5)  Θ_{0} = (∂_{ξ_{0}}h_{j,2})^{−1}g_{j,0} dh_{j,2}, on \overline{Γ_{j}}, 1 ⩽ j ⩽ p

and z_{0} = 1, the residues calculus made in the proof of part A implies that

∫_{∂Γ} z_{ℓ}φ_{η}^{−1}θ_{0} = ∑_{1⩽j⩽p} g_{j,0}h_{j,ℓ} , 1 ⩽ j ⩽ p.

Hence, ∫γ̃ φ_{η}θ_{ℓ} = 0. As γ̃ is contained in the affine space E_{ξ_{∗}} = ℂℙ_{2}\Λ_{ξ_{∗}}, we can apply [20, cor. 4.2 p. 265] and [11, prop. 1] and get in E_{ξ_{∗}}\γ̃ a complex curve X̃ of finite volume where θ_{0} extends weakly in a weakly holomorphic form Θ_{0} satisfying^{(}^{14}^{)}

(5.6)  ∫_{X} ( \overline{∂}φ ) ∧ Θ_{0} = ∫_{X} d (φΘ_{0}) = ∫_{γ} φθ_{0}

holds for any φ smooth in a neighborhood of \overline{X} and analytic near σ.

By construction, X = X̃ ∪ \overline{Γ} is a complex curve of ℂℙ_{2}\γ where θ_{0} extends as a weakly holomorphic form Θ_{0}, this extension coinciding in Γ with the form defined by (5.5). If ℓ = 1, 2, the form Θ_{ℓ} = z_{ℓ}Θ_{0} is a weakly holomorphic extension of θ_{ℓ} to X.

Let ε > 0 and let W_{ε} an ε-neighborhood of Λ_{ξ_{∗}}. As X_{ε} = X \\overline{W_{ε}} lies in the affine space E_{ξ_{∗}}\W_{ε}, [18, th. 4.7] implies that

(5.7)  d [X_{ε}] = n_{ε} [γ] + Σ_{1⩽j⩽p}n_{ε,j} [γ_{ε,j}]

where for 1 ⩽ j ⩽ p, n_{ε}, n_{ε,j} ∈ ℤ and γ_{ε,j} is the (smooth) boundary of the smooth (manifold) W_{ε} ∩ Γ_{j}. If 0 < ε′ < ε,

Σ_{1⩽j⩽p}d [Γ_{j} ∩ (W_{ε}\W_{ε}′)] = −d [X_{ε}] + d [X_{ε}′]
= (−n_{ε} + n_{ε}′) [γ] + Σ_{1⩽j⩽p} (−n_{ε,j} + n_{ε′,j}) [γ_{ε,j}]

Hence, n_{ε} = n_{ε}′ =^{def} n and as each Γ_{j} ∩ W_{ε}\W_{ε}′ is a smooth manifold with boundary, n_{ε,j} = n_{ε′,j} = −1. Taking now limits in (5.7) when ε goes to zero, we get d [X] = n [γ]. We suppress from X any compact component it may have and still denote the result by X ; note that X has now to be

\frac{1}{2πi} ∫_{γ} z_{1}^{m} \frac{θ_{0}}{ξ_{0} + ξ_{1}z_{1} + z_{2}} = Σ_{1⩽j⩽p}h_{j}^{m} (ξ) g_{j,0} (ξ)

14In [20][11], (5.6) is in fact obtained only for φ smooth on X and holomorphic in a neighborhood of γ but (5.6) follows from this together with (5.4) and the residue relations (very close in spirit to the relations T_{m,ξ})

where h_{j} (ξ), h_{j,2} (ξ) and g_{j,0} (ξ) are as above ; indeed these relations enable for generic ξ the computation of g_{j,0} (ξ) by a kramerian system and hence imply the smoothness of Θ_{0} near points of γ\σ. However, this precision is used not essentialy in the sequel.

connected. Since X is a complex curve of ℂℙ_{2}\γ, [10] implies that if n ≠ 0, there is in γ a compact set σ such that h^{1} (σ) = 0 and ( \overline{X} , ±γ ) is manifold with boundary near points of γ\σ ; as X is connected, this implies n = ±1. When n = 0, the structure theorems of Harvey-Shiffman [17] implies that Z =\overline{X} is then a complex compact curve of ℂℙ_{2} ; since γ is smooth, γ is locally a Jordan curve of Z_{reg} and the points where γ may meet the finite set Z_{sing} are only self-intersection points of Z.

Since X ∩ Λ_{ξ} = {(1 : h_{j} (ξ) : h_{j,2} (ξ)) , 1 ⩽ j ⩽ p}, the Stokes’s formula gives G = Σh_{j} (see proof of theorem 3a.A).

Assume that (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} ≠ 0. Then n ≠ 0 because otherwise for ξ closed to ξ_{∗}, the intersection of X with the line Λ_{ξ} should have to be the intersection with Λ_{ξ} of a compact Riemann surface, namely \overline{X}, which by a theorem of Reiss would force Σh_{j} to be affine in ξ_{0} (see [16, ch. 5.2] or [20, section 2]). Reasoning like in the proof of 3a.B, we also eliminate the possibility d [X] = − [γ] because it would imply that if H = {h_{1}, ..., h_{p}} is minimal in the sense that no proper subset of H satisfy (a1), each h_{j} also satisfies h_{j,y} = −h_{j,x}h_{j}. Hence n = 1.

When (∂^{2}G/∂ξ_{0}^{2})_{|W_{ξ_{∗}}} = 0, that is when Σh_{j} is affine in ξ_{0}, and when \overline{X} is not an algebraic curve where γ is a slit, n has to be non null and hence is ±1. Since G̃ |_{W_{η_{∗}}} ≠ 0, the minimal H in the above sense is not empty and reasoning likewise, we get d [X] = [γ].

Case G̃ |_{W_{η_{∗}}} = 0. This means that the minimal H is empty. Then, we can apply [20, th. 4.2 p. 264] in the affine case (see [11] for a generalization and a detailed proof in this case) to get in ℂ^{2}\γ (here ℂ^{2} is the complement of Λ_{ξ_{∗}} = {ξ_{∗0}w_{0} + ξ_{1∗}w_{1} + w_{2} = 0} in ℂℙ_{2}) a complex curve X of finite volume where θ_{0} extends weakly in a weakly holomorphic form Θ_{0} satisfying (5.6).

Since θ_{0} yields a non zero measure on γ which, because G̃ |_{W_{η_{∗}}} = 0, is orthogonal to all polynomials of ℂ^{2} ∼ ℂℙ_{2}\Λ_{ξ_{∗}}, we can apply Bishop [7] and [26] (Wermer originates and solves this problem [29] for the real analytic case) to get that X is the polynomial hull γ̃ of γ in ℂ^{2} and d (± [X]) = [γ].

[18] (see also [10]) implies then that γ contains a compact set σ such that h^{1} (σ) = 0 and ( \overline{X} , γ ) is manifold with boundary near points of γ\σ.

To prove (b2), we go back to the assumption that (a1) is true and we assume in addition that Θ_{ℓ} is holomorphic and satisfies (1.8) for all c ∈ H_{1} (X_{reg}). Then, there exists U_{ℓ} ∈ C^{∞} (X_{reg}) such that Θ_{ℓ} = dV_{ℓ} on X_{reg} ; since Θ_{ℓ} is a (1,0)-form, ∂V_{ℓ} = dV_{ℓ} = Θ_{ℓ}. We know from the preceding point that d [X] = n [γ] where n ∈ {0, 1}, up to a change of orientation of γ when G̃ |_{W_{η_{∗}}} = 0. Assume at first that n = 1. As d is elliptic up to the boundary, V_{ℓ} has to be smooth up to the boundary in the classical sense near points of γ outside σ and the preceding equality (5.6) yields dv_{ℓ} = du_{ℓ} where v_{ℓ} = V_{ℓ} |_{γ}. Hence, z_{0} is a point of γ where ( \overline{X} , γ ) is a manifold with boundary, there is a constant c such that v_{ℓ} = u_{ℓ} + c near z_{0} in γ. Since h^{1} (σ) = 0 and dv_{ℓ} = du_{ℓ} is smooth we have v_{ℓ} (z) − v_{ℓ} (z_{0}) = ∫_{γ_{z_{0},z}} du_{ℓ} where γ_{z_{0},z} is the positively oriented path of γ starting at z_{0} and ending at z. Hence, v_{ℓ} (z) − u_{ℓ} (z_{0}) − c = u_{ℓ} (z) − u_{ℓ} (z_{0}) and v_{ℓ} (z) = u_{ℓ} (z) + c. This implies that U_{ℓ} is a weakly harmonic extension of u_{ℓ}. When n = 0, \overline{X} is two sided locally near points of γ and each local side has the same boundary regularity as in the case n = 1. Hence, we can reason as in this case and get that there is a weakly harmonic extension of u_{ℓ} to X.

When we assume also that ∫_{X_{reg}} Θ_{ℓ} ∧ \overline{Θ_{ℓ}} < +∞, Θ_{ℓ} is holomorphic in the sense that its pullback to any normalization π : Z → X of X is holomorphic and not only meromorphic. The isolated singularities that the pullback V_{ℓ} of U_{ℓ} may have in Z are removable because dV_{ℓ} is smooth. So U_{ℓ} is harmonic on X and U_{ℓ} is the harmonic extension of u_{ℓ} to X. The proof is complete.

5.6. Proof of theorem 3b. The map f enable us to embed the abstract IDN-problem of theorem 3b in the projective but concrete frame of theorem 3c. So theorem 3b.A is a direct consequence of 3c.A.

For the converse part B.b1, we apply 3c.B to δ = f (γ) and θ_{ℓ} = θu_{ℓ}, ℓ = 0, 1, 2. We get in ℂℙ_{2}\δ an irreducible complex curve Y such that d [Y] = n [δ] where n ∈ {0, 1}, up to a change of orientation when G̃ |_{W_{η_{∗}}} = 0 ; in addition, each θ_{ℓ} extends weakly to Y into a weakly holomorphic (1,0)-form Θ_{ℓ}^{Y}.

When n = 1, the boundary regularity of Y mentioned in the proof of theorem 3c.B enable to apply readily the construction made in the proof of theorem 3a : adding to Y a subset σ′ of γ of zero one dimensional Hausdorff measure, we get an abstract complex curve Ỹ which can be normalized in the classical sense into an abstract Riemann surface X ; γ can then be topologically glued to X so that ( \overline{X} , γ ) becomes a manifold with almost smooth boundary where the pullback F to X of the meromorphic map ℂℙ_{2} ∋ z ↦ (z_{1}, z_{2}) gives a meromorphic extension of f to X. Since the forms Θ_{ℓ}^{Y} are meromorphic on Y, Θ_{ℓ} = F^{∗}Θ_{ℓ}^{Y} is well defined and meromorphic outside X \F^{−1}(σ′) which has zero length.

When n = 0, Z = \overline{X} is an algebraic curve and one can use a standard normalization of Z to get the same kind of conclusions.

The supplementary hypothesis of part B.b2 force each Θ_{ℓ} to have only removable singularities. Reasoning like in the proof of 3c.B.b2, (1.8) implies that Θ_{ℓ} = dV_{ℓ} = ∂V_{ℓ} for some harmonic function V_{ℓ} smooth up to regular boundary points of \overline{X} (when n = 0, γ cuts locally X into two domains and this means that each restriction of V_{ℓ} to these domains is smooth up to γ) and that there is a constant c such that U_{ℓ} + c agrees with u_{ℓ} on γ._{□}

6. Characterizations, effective or affine

6.1. Explicit integro-differential characterization. In this section where th. 4 is proved, (ξ_{0}, ξ_{1}) is replaced by the simpler (x, y) and we reason in a neighborhood of (0, 0). O_{0} is the set of one variable holomorphic functions near 0 is denoted by, O_{0} ⊗ ℂ_{d} [Z] stands for the set of polynomials of degree at most d with independent variable Z and coefficients in O_{0}. An element of O_{0} ⊗ ℂ_{d} [X] (resp. O_{0} ⊗ ℂ_{d} [Y]) should be think as a function of the type

(x, y) ↦ Σ_{0⩽j⩽d}λ_{j} (x) y^{j} (resp. (x, y) ↦ Σ_{0⩽j⩽d}λ_{j} (y) x^{j}) where each λ_{j} ∈ O_{0}.

If h is differentiable, the derivative of h with respect to one of its variable u is denoted h_{u}. If U is an open set in ℂ^{n}, O (U) is the space of holomorphic functions in U ; if h ∈ O (U) and 0 ∈ U, we set h_{x^{−0},y^{0}} = h and in any simply connected neighborhood of 0 in U, we denote by h_{x^{−α−1},y^{β}} (α, β ∈ ℕ) the function which vanishes at 0 and satisfies ∂ (h_{x^{−α−1},y^{β}}) /∂x = h_{x^{−α},y^{β}} ; h_{x,^{α}y^{−β−1}} is defined similarly.

A two variable function h is called a shock wave function on a domain D of ℂ^{2} if it is holomorphic and satisfies h_{y} = h_{x}h on D.

A p-algebroïde function on D is a p-uple h = (h_{1}, ..., h_{m}) of functions from D to ℂ for which one can find p holomorphic functions a_{0}, , ..., a_{p−1} in D such that for z ∈ D, h_{1} (z) , ..., h_{p} (z) are the roots, with multiplicities, of the polynomial T^{h} = X^{p} + a_{p−1} (z) X^{p−1} + ⋯ + a_{1} (z) X + a_{0} (z).

A p-multivaluate shock wave function on D is a p-algebroïde function h on D such that Σ = {Discr T^{h} = 0} is a hypersurface of D and for any z_{∗} ∈ D\Σ, the holomorphic functions h_{1}^{∗}, .., h_{p}^{∗} which near z_{∗} describe the roots of T_{z}^{h} are non null shock wave functions ; the first symmetric function of T^{h}, that is the sum of the roots of T^{h}, is called the trace of T^{h} of h. Traces of p-multivaluate shock wave functions are called p-shock wave functions.

Lemma 16. Consider p mutually distinct functions h_{1}, ..., h_{p} holomorphic in a domain D of ℂ^{2}. Then, each h_{j} is a shock wave function if and only the functions σ_{k} = (−1)^{k} Σ_{1⩽j_{1}<···<j_{k}⩽p}h_{j_{1}}...h_{j_{k}} satisfy the following system

(6.1) σ_{p}σ_{1,x} + σ_{p,y} = 0 & σ_{k}σ_{1,x} + σ_{k,y} = σ_{k+1,x} , 1 ⩽ k ⩽ p − 1.

Proof. Set T = X^{p} + Σ_{1⩽k⩽p}σ_{k}X^{p−k} and T′ = ∂T/∂X. The relations 0 =

(T h)_{x} = (T h)_{y} and h^{p} = − Σ_{1⩽k⩽p}σ_{k}h^{p−k} yields (T′h) (h_{y} − hh_{x}) = Sh with

S = Σ_{1⩽k⩽p−1} [σ_{k+1,x} − σ_{k}σ_{1,x} − σ_{k,y}] X^{p−k} − (σ_{p}σ_{1,x} + σ_{p,y})

Since deg S ⩽ p − 1, the fact that each h_{j} is a shock wave function implies that the coefficients of S vanish in a non empty open set and thus in the domain D. If S = 0 at every point of D, then each h_{j} verifies h_{y} − h_{x}h = 0 in the domain D because T′h_{j} ≢ 0. □

Proposition 17. Let D be a simply connected domain of ℂ^{2} containing 0, Δ its image by the projection (x, y) ↦ y and and H ∈ O (D). When u is differentiable, we set

D_{H}u = e^{H_{x,y^{−1}}} ∂(ue^{−H_{x,y^{−1}}} )/∂y & L_{H}u = (D_{H}u)_{x^{−1}}

The following two assertions are equivalent 1. H is a p-shock wave function in D.

2. There exists λ_{1}, ..., λ_{p−1} ∈ O (Δ) such that for λ̃_{j} (x, y) = λ_{j} (y), 1 ⩽ j ⩽ p − 1,

D_{H}L_{H}^{p−1}H = D_{H}L_{H}^{p−2}λ̃_{1} + ⋯ + D_{H}L_{H}^{0} λ̃_{p−1} (6.2) Discr T_{z} ≢ 0 (6.3)

where T_{z} = X^{p} + Σ_{1⩽k⩽p}s_{k} (z) X^{p−k} with

(6.4)  s_{k} = −L_{H}^{k−1}H + L_{H}^{k−2}λ̃_{1} + ⋯ + L_{H}^{0} λ̃_{k−1}, 1 ⩽ k ⩽ p.

More precisely, in case (2) is true, T determines a p-multivaluate shock wave function with trace H. Conversely, if H is the trace of a p-multivaluate shock wave function T, the p holomorphic functions which near a point z_{∗} in {Discr T ≠ 0} describes the roots of T_{z} have symmetric functions (−1)^{k} s_{k},1 ⩽ k ⩽ p which satisfy (6.2) and (6.4).

Proof. 1) Assume H is a p-shock wave function in D. Then, H is the first symmetric function of some T ∈ O (D)⊗ℂ_{p} [Z], Σ = {Discr T = 0} ≠ D and for any fixed z_{∗} ∈ D\Σ, deg T_{z_{∗}} = p and the holomorphic functions h_{1}^{∗}, .., h_{p}^{∗} which near z_{∗} describe the roots of T_{z} are non null shock wave functions with no common value on a sufficient small convex neighborhood W = U × V of z_{∗}. For k ∈ {1, ..., p} and on W, set ρ_{k} = σ_{k}e^{−H_{x,y^{−1}}} where σ_{k} is defined in lemma 16. Then (6.1) implies that ρ_{p,y} = 0 and that for k ∈ {1, ..., p − 1},

(ρ_{k+1}e^{H_{x,y^{−1}}} )_{x} = −H_{x}σ_{k} + σ_{k,y} = e^{H_{x,y^{−1}}} \frac{∂}{∂y} σ_{k}e^{−H_{x,y^{−1}}} = e^{H_{x,y^{−1}}} ρ_{k,y}

which yields λ_{k} ∈ O (V) such that

ρ_{k+1} (x, y) e^{H_{x,y^{−1}}} = [e^{H_{x,y^{−1}}} ρ_{k,y}]_{x^{−1}} + λ_{k} (y) .

Since σ_{1} = −H, we get e^{−H_{x,y^{−1}}} [e^{H_{x,y^{−1}}} ρ_{1,y}]_{x^{−1}} = −e^{−H_{x,y^{−1}}} L_{H}H. Setting

λ̃_{0} = −H, we obtain ρ_{2} = e^{−H_{x,y^{−1}}} (L_{H}λ_{0} + λ̃_{1}) and a straightforward finite recurrence gives (6.4) with (s_{k}) = (σ_{k}). In particular, k = p yields (6.2), because ρ_{p,y} = 0 ; the discriminant of T_{z} don’t vanish in W because h_{1}, ..., h_{p} have no common value. Since (6.4) also reads σ_{k+1} = L_{H}σ_{k} + λ̃_{k} we obtain that λ_{k} = σ_{k+1} (0, .) on V. Hence, λ_{1}, ..., λ_{p} do not depend on z_{∗} so that they are well defined holomorphic function on Δ.

2) Assume now (2) is true. We only have to check that T = X^{p} − Σ_{1⩽k⩽p}s_{k}X^{p−k} is actually a p-multivaluate shock wave function. Formulas

(6.4) also read s_{k+1} = L_{H}s_{k} + λ̃_{k} for 1 ⩽ k ⩽ p − 1 and (6.2) means that D_{H}s_{p} = 0. Hence 0 = s_{p,y} − H_{x}s_{p} = s_{p,y} − s_{1,x}s_{p} and s_{k+1,x} = D_{H}s_{k} = s_{k,y} − H_{x}s_{k} = s_{k,y} + s_{1,x}s_{k}. So, if z_{∗} ∈ D is outside Σ = {Discr T = 0}, lemma 16 implies that the holomorphic functions h_{1}, ..., h_{p} which near z_{∗} describe the roots of T_{z} are mutually distinct shock wave function. □

The following describes p-shock wave functions which are affine in x.

Proposition 18. Let D be a simply connected domain of ℂ^{2} containing 0, a, b ∈ O (D) and H = x ⊗ a + 1 ⊗ b. Then, H is a p-shock wave function, if and only if there exists Q_{0}, Q_{1} ∈ ℂ_{p−1} [Y] such that

a = \frac{Q_{1}}{1 − Q_{1,y^{−1}}} & b = \frac{Q_{0}}{1 − Q_{1,y^{−1}}} (6.5)

Discr (1 − Q_{1,y^{−1}}) ≠ 0, (6.6)

In addition, when (6.5) and (6.6) are satisfied, the decomposition of H in elementary fractions gives H as a sum of shock wave functions.

Proof. Assume H = Σ_{1⩽j⩽p}h_{j} where h_{1}, ...,h_{p} are mutually distinct shock

wave functions ; with proposition 17 notations, (−1)^{1} s_{1}, ..., (−1)^{p} s_{p} are the symmetric functions of h_{1}, ...,h_{p} and satisfy the relations D_{H}s_{p} = 0 and s_{k+1} = L_{H}s_{k} + λ̃_{k}, 1 ⩽ k ⩽ p − 1. There exist sequences of holomorphic functions, (λ_{j,k})_{k∈ℕ}, (a_{k})_{k∈ℕ} and (b_{k})_{k∈ℕ}, each satisfying the recurrence u_{k+1} = u′_{k} − au_{k}, such that

L_{H}^{k} λ̃_{j} = \frac{x^{k}}{k!} ⊗ λ_{j,k} , L_{H}^{k}H = \frac{x^{k+1}}{(k + 1)!} ⊗ a_{k} + \frac{x^{k}}{k!} ⊗ b_{k} , k ∈ ℕ

Hence, (6.2) yields the vanishing of the x-polynomial

\frac{x^{p}}{p!} ⊗ a_{p} + \frac{x^{p−1}}{(p − 1)!} ⊗ b_{p} − ∑_{1⩽j⩽p−1} \frac{x^{p−1−j}}{(p − 1 − j)!} ⊗ λ_{j,p−j}

and ensures a_{p} = b_{p} = λ_{j,p−j}, 1 ⩽ j ⩽ p − j. So, one can find Q_{1}, Q_{0} ∈ ℂ_{p−1} [Y] and Λ_{j} ∈ ℂ_{p−j−1} [Y], 1 ⩽ j ⩽ p − 1, such that with A = a_{y^{−1}}

a_{k} = Q_{1}^{(k)}e^{A} , b_{k} = Q_{0}^{(k)}e^{A}, λ_{j,k} = Λ_{j}^{(k)}e^{A}, k ∈ ℕ.

Thus a = a_{0} = Q_{1}e^{A}, 1 − e^{−A} = Q_{1,y^{−1}} and hence, (6.5).

Conversely, assume that (6.5) and (6.6) are true for some Q_{0}, Q_{1} ∈ ℂ_{p−1} [Y]. Then, the decomposition in elementary fraction of H is Σh_{j} where h_{j} (x, y) = \frac{q_{j}x+c_{j}}{1−q_{j}y}, 1 ⩽ j ⩽ p and q_{1}^{−1}, ..., q_{p}^{−1} are the roots of 1 − Q_{1,y^{−1}}. It is quite evident that h_{1}, ..., h_{p} are mutually distinct shock wave functions. □

Remark 1. If x ⊗ a + 1 ⊗ b is the sum of p-mutually distinct shock wave functions h_{1}, ...,h_{p}, each h_{j} is algebraic since for 1 ⩽ k ⩽ p, (6.4) yields s_{k} (x, y) = [1 − Q_{1y^{−1}} (y)]^{−1} S_{k} (x, y) where S_{k} is the polynomial defined by

S_{k} = −\frac{x^{k} ⊗ Q_{1}^{(k−1)}}{k!} − \frac{x^{k−1} ⊗ Q_{0}^{(k−1)}}{(k − 1)!} + ∑_{1⩽j⩽k−1} \frac{x^{k−1−j} ⊗ Λ_{j}^{(k−1−j)}}{(k − 1 − j)!} .

Remark 2. When Q_{1} is allowed to be non generic, 1 − Q_{1,y^{−1}} can only

be written in the form 1 − Q_{1,y^{−1}} = Π_{1⩽j⩽m} (1 − q_{j}y)^{α^{j}} with α_{1}, ..., α_{p} ∈ ℕ^{∗}.

Hence, if Q_{0} ∈ ℂ_{p−1} [Y] and (a, b) is defined by (6.5), there is constants c_{j,ℓ} such that

H =^{def} x ⊗ a + 1 ⊗ b = ∑_{1⩽j⩽m} ∑_{1⩽ℓ⩽α_{j}} h_{j,ℓ}

where h_{j,ℓ} = \frac{q_{j}x}{1−q_{j}y} + \frac{c_{j,ℓ}}{(1−q_{j}y)^{ℓ}}, 1 ⩽ j ⩽ m, 1 ⩽ ℓ ⩽ α_{j}. Each h_{j,ℓ} is now a generalized shock wave function in the sense it is a solution of the equation

h_{y} − h_{x}h = (ℓ − 1) κh_{x}^{ℓ+1}

where κ ∈ ℂ is equal to c_{j,ℓ}/q_{j}^{ℓ}.

Generalized shock wave functions arise when an affine function H = x ⊗ a + 1 ⊗ b has coefficient a and b given by (6.5) not constraint to (6.6). In that case, H is clearly a limit of p-shock wave functions. The lemma below, which is an elementary consequence of prop. 18, proves the converse and so, shows that generalized shock wave functions occur naturally.

Lemma 19. Let (H^{t})_{t∈T} = (x ⊗ a^{t} + 1 ⊗ b^{t})_{t∈T} be a continuous family of holomorphic affine functions in a simply connected domain D such that the set T_{reg} of parameters t for which H_{t} is a p-shock wave function is dense in T, then there exists in ℂ_{p−1} [Y] continuous family of holomorphic polynomials

(Q_{1}^{t}) and (Q_{0}^{t}) such that for any t ∈ T, a^{t} = Q_{1}^{t} (1 − Q_{1,y^{−1}}^{t})^{−1} and b^{t} = Q_{1,0}^{t} (1 − Q_{1,y^{−1}}^{t})^{−1}. Hence, H^{t} is a p-shock (resp. p-generalized) shock

wave function when Discr (1 − Q_{1,y^{−1}}^{t}) ≠ 0 (resp. Discr (1 − Q_{1,y^{−1}}) = 0).

6.1.1. Proof of theorem 4. 1) Assume that there is an open Riemann surface X such that \overline{X} = X ∪ γ is a manifold with almost smooth boundary. Then, theorem 3a gives that almost all point (ξ_{0∗}, ξ_{1∗}, 1) of ℂℙ_{2} has a neighborhood W_{ξ_{∗}} for which one can find an integer p and p mutually distinct shock wave functions h_{1}, ..., h_{p} on W_{ξ_{∗}} such that L = Σh_{j} − G is affine in ξ_{0}.

Set s_{k} = (−1)^{k} Σ_{1⩽j_{1}<···<j_{k}⩽p}h_{j_{1}}...h_{j_{k}} and T = X^{p} + Σ_{1⩽k⩽p}s_{k}X^{p−k}. Then

Discr T ≢ 0 and proposition 17 implies that property (1) of th. 4 holds.

Assume now that p̃ is the least integer q such that there is a q-multivaluate shock wave function whose trace differs from G |_{W_{ξ_{∗}}} only by a ξ_{0}-affine

function. Let T̃ be a p̃-multivaluate shock wave function with trace H̃ such that L̃ = H̃−G |_{W_{ξ_{∗}}} is affine in ξ_{0}. Let h̃_{1}, ..., h̃p̃ be the holomorphic function

on W_{ξ_{∗}} which describes the roots of T̃. Then {h̃_{1}, ..., h̃p̃} is minimal in the sense of the fourth remark below theorem 3a.

When p̃ ⩾ 1, this remark says that X is a normalization of the analytic extension Y in ℂℙ_{2}\f (γ) of the union of the graphs of the functions (1 : h̃_{j} : −ξ_{0} − ξ_{1}h̃_{j}), 1 ⩽ j ⩽ p̃, and for any ξ ∈ W_{ξ_{∗}}, the intersection of Y with the projective lines ξ_{0}w_{0} + ξ_{1}w_{1} + w_{2} = 0 is

{(1 : h̃_{j} (ξ) : ξ_{0} − ξ_{1}h̃_{j} (ξ)) ; 1 ⩽ j ⩽ p̃} .

Since by prop 10 Y is uniquely determined by (γ, f), each h̃_{j} and so, each symmetric function s̃_{k} of h̃_{1}, ..., h̃_{k}, is uniquely determined by (γ, f). If λ_{1}, ..., λp̃ are any one variable holomorphic function such that assertion (2) of prop. 17 holds, with H̃ instead of H, then λ_{k−1} (ξ_{1}) = s̃_{k} (0, ξ_{1}), 1 ⩽ k ⩽ p − 1. Hence, λ_{1}, ..., λp̃ are uniquely determined by (γ, f).

Lemma 13 implies now that L is affine in ξ_{0} and is, for some integer q, the limit of a continuous one parameter family of ξ_{0}-affine q-shock wave functions. Thanks to lemma 19, this implies that L (ξ_{0}, ξ_{1}) = ξ_{0} \frac{α′(ξ_{1})}{1−α(ξ_{1})} +

\frac{β(ξ_{1})}{1−α(ξ_{1})} where α ∈ ℂ_{q} [ξ_{1}] vanish at zero and b ∈ ℂ_{q−1} [ξ_{1}].

2) Assume now that property (1) of th. 4 holds. Set T = X^{p}+ Σ_{1⩽k⩽p}s_{k}X^{p−k}

where now s_{k} are defined by (6.4). Proposition 17 implies then that T is determines a p-multivaluate shock wave function whose trace is G + L. Since L is affine in ξ_{0}, G satisfies the hypothesis of theorem 3a.B. As G is not affine in ξ_{0} by hypothesis, γ, with its given orientation, is the boundary of an open Riemann surface X with the sought properties. Hence, (4) has to be satisfied from the direct part of theorem 4.

6.1.2. Particular case of theorem 4. For minimal p equal to 2, it turns out that theorem 4 says that γ bounds almost smoothly an open Riemann surface X where f extends meromorphically if and only if for some constants c, α_{1}, α_{2}, β_{0}, β_{1}

(6.7)  D_{G}L_{G}G = cC + α_{1}A_{1} + α_{2}A_{2} + β_{0}B_{0} + β_{1}B_{1}

where

C = (G_{x} − g_{x}) e^{g_{x,y^{−1}}}
A_{1} = yD_{G}L_{G}G + (xL_{G}G)_{x} , A_{2} = y^{2}D_{G}L_{G}G + (2xyL_{G}G + x^{2}G)_{x}
B_{0} = D_{G}G − D_{G}g , B_{1} = yB_{0} + g − G

6.2. Affine characterization. A characterization for an affine presentation is possible but very special data have to be selected. Assume X is a Rieman surface with almost smooth boundary γ. When u ∈ C^{∞} (γ) a straightforward computation gives that d^{c}ũ = (Nu) τ^{∗} on γ\\overline{X}_{sing}. So, lemma 11 implies that

(6.8)  ∫_{γ} (Nu) τ^{∗} = ∫_{X} dd^{c}ũ = 0.

Hence (Nu) τ^{∗} has a primitive v on γ and the holomorphic extension to X of h = u + iv is equivalent to the moments condition

(6.9)  ∀Ψ ∈ H^{1,0}(\overline{X}), ∫_{γ} hΨ = 0.

If H ∈ O( \overline{X} )^{N+1} is such that h = H |_{γ} embeds γ in ℂ^{N+1}, Y = H (X) \h (γ) is a complex curve of ℂ^{N+1}\h (γ) which has finite volume and has boundary h (γ) and γ has to satisfy the Harvey-Lawson-moment conditions

(6.10)  ∀k_{0}, ..., k_{N} ∈ ℕ, ∫_{γ} h_{0}^{k_{0}}...h_{N}^{k_{N}} dh_{0} = 0

which by the way contains also (6.8).

Proposition 20. Let u_{0}, ..., u_{N} ∈ C^{∞} (γ) and let v_{ℓ} ∈ C^{∞} (γ) be a primitive of (Nu_{ℓ}) τ^{∗}, 0 ⩽ ℓ ⩽ N. Assume that h = (u_{ℓ} + iv_{ℓ})_{0⩽ℓ⩽N} is an embedding of γ in ℂ^{N+1}. Then (6.10) is a necessary condition to the existence of a Riemann surface X with almost smooth boundary γ such that each u_{ℓ} extends to X as a harmonic function with harmonic conjugate function. The converse is true when γ is connected and suitably oriented.

Remarks. Of course, the above conclusion means that X is a Riemann surface where each h_{ℓ} extends holomorphically.

Proof. If X exists with the required properties, δ = h (γ) bounds ,in the sense of current, h (X) \δ which is a complex curve of finite volume of ℂ^{n}\δ. Cauchy theorem implies then that (6.10) is verified. If (6.10) is satisfied, [18] produces a holomorphic 1-chain Y such that dY = [δ] ; since γ is connected, Y = [Y] for a suitable orientation of γ. A normalization of \overline{Y} constructed as in the proof of th. 3a gives a suitable X. □

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(G. Henkin and V. Michel) Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05

E-mail address: henkin@math.jussieu.fr, michel@math.jussieu.fr