A MORSE-THEORETICAL PROOF OF THE HARTOGS EXTENSION THEOREM

JO ËL MERKER AND EGMONT PORTEN ABSTRACT. 100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: holomorphic functions in a connected neighborhood V(∂Ω) of a connected boundary ∂Ω ⋐ C n (n 2) do extend holomorphically and uniquely to the domain Ω. Martinelli in the early 1940's and Ehrenpreis in 1961 obtained a rigorous proof, using a new multidimensional integral kernel or a short ∂ argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), [START_REF] Hartogs | Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen[END_REF] and E.E. [START_REF] Levi | Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse[END_REF] in some special, model cases. In fact, known attempts (e.g. [START_REF] Osgood | Lehrbuch der Funktionentheorie[END_REF][START_REF] Brown | On certain analytic continuations and analytic homeomorphisms[END_REF]) struggled for monodromy against multivaluations, but failed to get the general global theorem.

Moreover, quite unexpectedly, Fornaess in 1998 exhibited a topologically strange (nonpseudoconvex) domain Ω F ⊂ C 2 that cannot be filled in by holomorphic discs, when one makes the additional requirement that discs must all lie entirely inside Ω F . However, one should point out that the standard, unrestricted disc method usually allows discs to go outsise the domain (just think of Levi pseudoconcavity).

Using the method of analytic discs for local extensional steps and Morsetheoretical tools for the global topological control of monodromy, we show that the Hartogs extension theorem can be established in such a way. [22 colored illustrations] §1. THE HARTOGS EXTENSION THEOREM AND THE METHOD OF ANALYTIC DISCS 100 years ago exactly, in 1906, the publication of Hartogs's thesis ( [START_REF] Hartogs | Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen[END_REF] under the direction of Hurwitz) revealed what is now considered to be the most striking fact of multidimensional complex analysis: the automatic, compulsory holomorphic extension of functions of several complex variables to larger domains, especially for a class of "pot-looking" domains, nowadays called Hartogs figures, that may be filled in up to their top. Soon after, E.E. Levi [START_REF] Levi | Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse[END_REF] applied the Hurwitz-Hartogs argument of Cauchy integration on complex affine Date: 2007-4- 26. circles moving in the domain (firstly discovered in [START_REF] Hurwitz | Über die Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit[END_REF]), in order to perform local holomorphic extension across strictly (Levi) pseudoconcave boundaries. The so-called method of analytic discs was born, historically.

Hartogs extension theorem.

Let Ω ⋐ C n be a bounded domain having connected boundary. If n 2, every function holomorphic in some connected open neighborhood V(∂Ω) of ∂Ω extend holomorphically and uniquely inside Ω, i.e.:

∀ f ∈ O V(∂Ω) , ∃ ! F ∈ O Ω ∪ V(∂Ω) s.t. F V(∂Ω) = f.
Classically, one also presents an alternative formulation, which is checked to be equivalent -think that K = Ω V(∂Ω).

Hartogs theorem bis . If

Ω ⋐ C n (n
2) is a domain and if K ⊂ Ω is any compact such that Ω\K connected, then O(Ω\K) = O(Ω) Ω\K .

Already in [START_REF] Hartogs | Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen[END_REF] (p. 231), Hartogs stated such a global theorem in the typical language of those days, without claiming single-valuedness howeversomething that he consistently mentions in other places. Later in [START_REF] Osgood | Lehrbuch der Funktionentheorie[END_REF], Osgood (who gives the reference to Hartogs) "proves" unique holomorphic extension with discs, but what is written there is seriously erroneous, even when applied to a ball. In 1936, well before Milnor ([31]) had popularized Morse theory, using topological concepts and a language which are nowadays difficult to grasp, Brown ([5]) fixed somehow single-valuedness of the extension 1 : discretizing Ω\K to tame the topology, he exhausts C n by spheres of decreasing radius (as we will do in this paper), but we believe that his proof still contains imprecisions, because the subtracting process that we encounter unavoidably when applying Morse theory with the same spheres does not appear in [START_REF] Brown | On certain analytic continuations and analytic homeomorphisms[END_REF].

Since the 1940's, few complex analysts have seriously thought about testing the limit of the disc method probably because the motivation was gone, and in fact, the possible existence of an elementary rigorous proof of the global Hartogs extension theorem using only a finite number of Hartogs figures remained a folklore belief; for instance, in [START_REF] Struppa | The first eighty years of Hartogs' theorem[END_REF], p. 133, it is just left as an "exercise". But to the authors' knowledge, no reliable mathematical publication shows fully how to perform a rigorous proof of the global theorem, using only the original Hurwitz-Hartogs-Levi analytic discs as a tool.

On the other hand, thanks to the contributions of Fueter ( [START_REF] Fueter | Über einen Hartogs'schen Satz[END_REF]), of Martinelli ( [START_REF] Martinelli | Alcuni teoremi integrali per le funzioni per le funzioni analitiche di più variabili complesse[END_REF][START_REF] Martinelli | Sopra una dimostrazione di R. Fueter per un teorema di Hartogs[END_REF]), of Bochner ([4]) and of Fichera ([9]), powerful multidimensional integral kernels were discovered that provided a complete proof, from the side of Analysis. Soon after, Ehrenpreis ( [START_REF] Ehrenpreis | A new proof and an extension of Hartogs' theorem[END_REF]) found what is known to be the most concise proof, based on the vanishing of ∂-cohomology with compact support. This proof was learnt by generations of complex analysts, thanks to Hörmander's book [START_REF] Örmander | An Introduction to Complex Analysis in Several Variables[END_REF]. Range's Correction of the Historical Record [START_REF] Range | Extension phenomena in multidimensional complex analysis: correction of the historical record[END_REF] provides an excellent account of the very birth of integral formulas in C n . Since the 1960's, ∂ techniques, L 2 methods and integral kernels developed into a vast field of research in Several Complex Variables, c.f. [START_REF] Örmander | An Introduction to Complex Analysis in Several Variables[END_REF][START_REF] Andreotti | Levi convexity and the Hans Lewy problem, I and II[END_REF][START_REF] Henkin | Theory of functions on complex manifolds[END_REF][START_REF] Henkin | The method of integral representations in complex analysis (Russian), Current problems in mathematics[END_REF][START_REF] Range | Holomorphic functions and integral representations in several complex variables[END_REF][START_REF] Chirka | Removable singularities in the boundary[END_REF][START_REF] Demailly | Analytic and algebraic geometry[END_REF][START_REF] Jarnicki | Extension of holomorphic functions[END_REF][START_REF] Öricke | Hulls and analytic extension from nonpseudoconvex boundaries[END_REF][START_REF] Lieb | The Cauchy-Riemann complex. Integral formulae and Neumann problem[END_REF][START_REF] Henkin | Principe de Hartogs dans les variétés CR[END_REF].

A decade ago, Fornaess [START_REF] Fornaess | The disc method[END_REF] produced a topologically strange domain Ω F that cannot be filled in by means of analytic discs, when one makes the additional requirement that discs must all lie entirely inside the domain. Possibly, one could interpret this example as a "defeat" of geometrical methods.

But in absence of pseudoconvexity, it is much more natural to allow discs to go outside the domain, because the local E.E. Levi extension theorem already needs that. In fact, as remarked by Bedford in his review [START_REF] Bedford | Review of [Fo1998[END_REF] of [START_REF] Fornaess | The disc method[END_REF], Hartogs' phenomenon for Fornaess' domain Ω F may be shown to hold straightforwardly by means of the usual, unrestricted disk method.

Furthermore, the study of envelopes of holomorphy (see the monograph of Jarnicki and Pflug [START_REF] Jarnicki | Extension of holomorphic functions[END_REF] for an introduction to Riemann domains spread over C n and [START_REF] Merker | Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities[END_REF] for applications in a CR context) shows well how natural it is to deal with sucessively enlarged (Riemann) domains. Bishop's constructive approach, especially his famous idea of gluing discs to real submanifolds, reveals to be adequate in such a widely open field of research. We hence may hope that, after the very grounding historical theorem of Hartogs has enjoyed a renewed proof, geometrical methods will undergo further developments, especially to devise fine holomorphic extension theorems that are unreachable by means of contemporary ∂ techniques.

In this paper, we establish rigorously that the Hartogs extension theorem can be proved by means of a finite number of parameterized families of analytic discs (Theorems 2.7 and 5.4). The discs we use are all (tiny) pieces of complex lines in C n . The main difficulty is topological and we use the Morse machinery to tame multisheetedness.

At first, we shall replace the boundary ∂Ω by a C ∞ connected oriented hypersurface M ⋐ C n (n 2) for which the restriction to M of the Euclidean norm function z → ||z|| is a good Morse function (Lemma 3.3), namely there exist only finitely many points

p λ ∈ M, 1 λ κ, with || p 1 || < • • • < || p κ || at which z → ||z||
restricted to M has vanishing differential. We also replace V(∂Ω) by a very thin tubular neighborhood V δ (M), 0 < δ < < 1, and Ω by a domain Ω M ⋐ C n bounded by M. Next, we will introduce a modification of the Hartogs figure, called a Levi-Hartogs figure, which is more appropriate to produce holomorphic extension from the cut out domains ||z|| > r ∩ Ω M , where the radius r will decrease, inductively. Local Levi pseudoconcavity of the exterior of a ball then enables us to prolong the holomorphic functions to ||z|| > rη ∩ Ω M , for some uniform η with 0 < η < < 1, which depends on the dimension n 2, on δ, and on the diameter of Ω. We hence descend stepwise to lower radii until the domain is fully filled in. However, this naive conclusion fails because of multivaluations and a crucial three-piece topological device is required. We begin by filling the top of the domain, which is simply diffeomorphic to a cut out piece of ball. Geometrically speaking, Morse points p λ , 1 λ κ, are the only points of M at which the family of spheres ||z|| = r 0<r<∞ are tangent to M. We denote || p λ || =: r λ with r 1 < • • • < r κ . In Figure 1, we have κ = 6. For an arbitrary fixed radius r with r λ < r < r λ+1 , and some fixed λ with 1 λ κ -1, we consider all connected components M c >r , 1 c c λ , of the cut out hypersurface M ∩ ||z|| > r . Their number c λ is the same for all r ∈ r λ , r λ+1 . In Figure 1, when r 3 < r < r 4 , we see three such components.

By descending discrete induction r → r-η, we show that each such connected hypersurface M c >r ⊂ ||z|| > r bounds a certain domain Ω c >r ⊂ ||z|| > r which is relatively compact in C n and that holomorphic functions in V δ (M) do extend holomorphically and uniquely to Ω c >r . While approaching a lower Morse point, three different topological processes will occur 2 : creating a new component Ω c ′ >r-η to be filled in further; merging two components Ω >r-η . The unavoidable multivaluation phenomenon will be tamed by the idea of separating ab initio the components M c >r , 1 c c λ . Indeed, an advantageous topological property will be shown to be inherited through the induction r → rη, hence always true, namely that two different domains Ω c 1 >r and Ω c 1 >r are either disjoint or one is contained in the other. Consequently, the multivaluation aspect will only happen in the sense that the two uniquely defined and univalent holomorphic extensions

f c 1 r to Ω c 1 >r and f c 2 r to Ω c 2 >r can be different on Ω c 1 >r , in case Ω c 1 >r ⊂ Ω c 2
>r , or vice versa. In this way, we avoid completely to deal with Riemann domains spread over C n . Some of the elements of our approach should be viewed in a broader context. In their celebrated paper [START_REF] Andreotti | Théorèmes de finitude pour la cohomologie des espaces complexes[END_REF] (see also [START_REF] Henkin | Andreotti-Grauert Theory by Integral Formulas[END_REF]), Andreotti and Grauert observed that convenient exhaustion functions can be used to prove very general extension and finiteness results on q-concave complex varieties. Their arguments implicitly contained a geometrical proof of the Hartogs extension theorem in the case 2 A certain number of other simpler cases will also happen, where the components Ω c >r do grow regularly with respect to holomorphic extension, possibly changing topology.

where the domain Ω ⊂ C n is pseudoconvex (whence Fornaess' counter-example must be nonpseudoconvex). However, in contrast to our finer method, the existence of an internal strongly pseudoconvex exhaustion function ρ on a complex manifold X excludes ab initio multisheetedness: indeed, in such a circumstance, extension holds stepwise from shells of the form z ∈ X : a < ρ(z) < b just to deeper shells {a ′ < ρ < b} with a ′ < a (details are provided in [START_REF] Merker | The Hartogs extension theorem on (n -1)-complete complex spaces[END_REF]), namely the topology is controlled in advance by ρ and multiple domains as Ω c >r above cannot at all appear.

There is a nice alternative approach to the (singular) Hartogs extension theorem via a global continuity principle, realized in [START_REF] Öricke | Hulls and analytic extension from nonpseudoconvex boundaries[END_REF] by Jöricke and the second author, with the purpose of understanding removable singularities by means of (geometric) envelopes of holomorphy. The idea is to perform holomorphic extensions along one-parameter families of holomorphic curves (not suppose to be discs). A basic extension theorem on some appropriate Levi flat 3-manifolds, called Hartogs manifolds in [START_REF] Öricke | Hulls and analytic extension from nonpseudoconvex boundaries[END_REF], is shown via stepwise extension in the direction of an increasing real parameter. The geometrical scheme of this construction has a common topological element with our method: the simultaneous holomorphic extension to collections of domains that are pairwise either disjoint or one is contained in the other.

On the other hand, our technique only rely upon the existence of appropriate exhaustion functions, without requiring neither the existence of Levi-flat 3-manifolds nor the existence of global holomorphic functions in the ambient complex manifold. In addition, inspired by a definition formulated by Fornaess in [START_REF] Fornaess | The disc method[END_REF], we establish that only a finite number of Levi-Hartogs figures is needed in the filling process. Finally, we would like to mention that a straightforward adaptation of the proof developed here would yield a geometrical proof of the Hartogs-type extension theorem of Andreotti and Hill ( [START_REF] Andreotti | Levi convexity and the Hans Lewy problem, I and II[END_REF]), which is valid for arbitrary domains in (n-1)-complete manifolds (in the sense of Andreotti-Grauert [START_REF] Grauert | Theory of q-convexity and q-concavity[END_REF]).

Twenty-two colored illustrations appear, each one being inserted at the appropriate place in the text. Abstract geometrical thought being intrinsically pictural, we hope to address to a broad audience of complex analysts and geometers. §2. PREPARATION OF THE BOUNDARY AND UNIQUE EXTENSION

Preparation of a good

C ∞ boundary. Denote by ||z|| := |z 1 | 2 + • • • + |z n | 2 1/2 the Euclidean norm of z = (z 1 , . . . , z n ) ∈ C n and by B n (p, δ) := ||z -p|| < δ the open ball of radius δ > 0 centered at a point p. If E ⊂ C n is any set, V δ (E) := ∪ p∈E B n (p, δ)
is a concrete open neighborhood of E.

As in the Hartogs theorem, assume that the domain Ω ⋐ C n has connected boundary ∂Ω and let V(∂Ω) be an open neighborhood of ∂Ω, also connected. Clearly, there exists δ 1 with 0 < δ 1 < < 1 such that ∂Ω ⊂ V δ 1 (∂Ω) ⊂ V(∂Ω); of course, V δ 1 (∂Ω) is then also connected. Choose a point p 0 ∈ C n with dist (p 0 , Ω) = 3, center the coordinates (z 1 , . . . , z n ) at p 0 and consider the distance function

(2.2) r(z) := ||z -p 0 || = ||z||.
It is crucial to prepare as follows the boundary, replacing (Ω, ∂Ω) by (Ω M , M), thanks to some transversality arguments that are standard in Morse theory ( [START_REF] Milnor | Morse Theory, Based on lectures notes[END_REF] and [START_REF] Hirsch | Differential topology[END_REF], Ch. 6).

Lemma 2.3.

There exists a C ∞ connected closed and oriented hypersurface M ⊂ V δ 1 /2 (∂Ω) such that: Sometimes, r M satisfying (ii) and (iii) is called a good Morse function on M. We will shortly say that M is a good boundary.

(i) M bounds a unique bounded domain Ω M with Ω ⊂ Ω M ∪ V(∂Ω); (
If k λ is the number of positive eigenvalues of the (symmetric) Hessian matrix H[r M ]( p λ ), the extrinsic Morse lemma ( [START_REF] Milnor | Morse Theory, Based on lectures notes[END_REF][START_REF] Hirsch | Differential topology[END_REF]) shows that there exist 2n real coordinates v, x 1 , . . . , x k λ , y 1 , . . . , y 2n-k λ -1 in a neighborhood of p λ in C n such that

• the sets {v(z) = cst} simply correspond 3 to the spheres {r(z) = cst} near p λ ; • x 1 , . . . , x k λ , y 1 , . . . , y 2n-k λ -1 provide (2n -1) local coordinates on the hypersurface M, whose graphed equation is normalized to be the simple hyperquadric

v = 1 j k λ x 2 j - 1 j 2n-k λ -1 y 2 j .
3

In fact, one can just take the translated radius r(z)r( p λ ) as the coordinate v = v(z).

Classically, the number (2nk λ -1) of negatives is called the Morse index of r(z) M at p λ ; we will call k λ its Morse coindex.

For rather general differential-geometric objects, Morse theory enables to control a significant part of homotopy groups and of (co)homologies, e.g. via Morse inequalities. In our case, we shall be able to control somehow the global topology of the cut-out domains Ω M ∩ {||z|| > r} that re external to closed balls of radius r, filling them progressively by means of analytic discs contained in small (Levi-)Hartogs figures (Section 3). We start by checking rigorously that the Hartogs theorem can be reduced to some good boundary. Given two connected open sets U 1 ⊂ C n and U 2 ⊂ C n with U 1 ∩ U 2 nonempty, we will say4 that O(U 1 ) extends holomorphically to U 1 ∪ U 2 if :

• the intersection U 1 ∩ U 2 is connected; • there exists an open nonempty set V ⊂ U 1 ∩ U 2 such that for every f 1 ∈ O(U 1 ), there exist f 2 ∈ O(U 2 ) with f 2 | V = f 1 | V .
It then follows from the principle of analytic continuation that

f 1 | U 1 ∩U 2 = f 2 | U 1 ∩U 2 ,
so that the joint function F , equal to f j on U j for j = 1, 2, is well defined, is holomorphic in U 1 ∪ U 2 and extends f 1 , namely

F | U 1 = f 1 .
In concrete extensional situations, the coincidence of f 1 with f 2 is controlled only in some small V ⊂ U 1 ∩ U 2 , so the connectedness of U 1 ∩ U 2 appears to be useful to insure monodromy. Sometimes also, we shall briefly write O(U 1 ) = O(U 1 ∪ U 2 ) U 1 , instead of spelling rigorously:

∀ f 1 ∈ O U 1 ∃ F ∈ O U 1 ∪ U 2 such that F U 1 = f 1 .
Lemma 2.6. Suppose that for some δ with 0 < δ δ 1 /2 so small that V δ (M) ≃ M ×(-δ, δ) is a thin tubular neighborhood of the good boundary M ⊂ V δ 1 /2 (∂Ω) ⊂ V(∂Ω), the Hartogs theorem holds for the pair (Ω M , V δ (M)): Letting p, q ∈ Ω M ∩ V(∂Ω), there exists a C ∞ curve γ : [0, 1] → V(∂Ω) connecting p to q. If γ meets M, let p ′ be the first point on γ ∩ M and let q ′ be the last one. We then modify γ, joining p ′ to q ′ by means of a curve µ entirely contained in the connected hypersurface M. It suffices to push µ slightly inside Ω M to get an appropriate curve running from p to q inside Ω M ∩ V(∂Ω). Thus,

O V δ (M) = O Ω M ∪ V δ (M) V δ (M ) .

Then the general Hartogs extension property holds:

O V(∂Ω) = O Ω ∪ V(∂Ω) V(∂Ω) . Proof. Let f ∈ O V(∂Ω) . By assumption, its restriction to V δ (M) ⊂ V(∂Ω) enjoys an extension F δ ∈ O Ω M ∪ V δ (M) . To ascertain that f and F δ coincide in Ω M ∩ V(∂Ω), connectedness of Ω M ∩ V(∂Ω) is welcome.
Ω M ∩ V(∂Ω) is connected. It follows, moreover, that the open set Ω M ∪ V δ (M) ∩ V(∂Ω) = Ω M ∩ V(∂Ω) V δ (M)
is also connected, so the coincidence

f = F δ , valid in V δ (M), propagates to Ω M ∩ V(∂Ω) ∪ V δ (M).
Finally, the function

F := F δ in Ω M ∪ V δ (M), f in V(∂Ω)\Ω M , is well defined since F δ = f in V δ (M)\Ω M ≃ M × (0, δ) , is holomorphic in Ω M ∪ V(∂Ω) = Ω ∪ V(∂Ω)
and coincides with f in V(∂Ω).

Thus, we are reduced to establish global holomorphic extension with some good, geometrically controlled data. Theorem 2.7. Let M ⋐ C n (n 2) be a connected C ∞ hypersurface bounding a domain Ω M ⋐ C n . Suppose to fix ideas that 2 dist 0, Ω M 5 and assume that the restriction r M := r| M of the distance function r(z) = ||z|| to M is a Morse function having only a finite number κ of critical points p λ ∈ M, 1 λ κ, located on different sphere levels:

2 r 1 := r( p 1 ) < • • • < r κ := r( p κ ) 5 + diam Ω M .
Then there exists δ 1 > 0 such that for every δ with 0 < δ δ 1 , the (tubular) neighborhood V δ (M) enjoys the global Hartogs extension property into Ω M :

O V δ (M) = O Ω M ∪ V δ (M) V δ (M ) ,
by "pushing" analytic discs inside a finite number of Levi Given ε ∈ R with 0 < ε < < 1 and a ∈ N with 1 a n -1, we split the coordinates z ∈ C n as (z 1 , . . . , z a ) together with (z a+1 , . . . , z n ), and we define the (na)-concave Hartogs figure by

H n-a ε := max 1 i a |z i | < 1, max a+1 j n |z j | < ε 1 -ε < max 1 i a |z i | < 1, max a+1 j n |z j | < 1 . 1 -ε 0 1 ε |z2| |z1| 1 H 2-1 ε z1 y2 C 2 ∆ 2 A ε ′ z 2 (∆) A ε ′ z 2 (∂∆)
H n-a ε := z ∈ C n : max 1 i n |z i | < 1 = ∆ n .
Proof. As in the diagram, we consider only n = 2, a = 1, the general case being similar. Pick an arbitrary

f ∈ O H 2-1 ε . Letting ε ′ with 0 < ε ′ < ε, letting z 2 ∈ C with |z 2 | < 1, the analytic disc ζ -→ [1 -ε ′ ] ζ, z 2 =: A ε ′ z 2 (ζ),
where ζ belongs to the closed unit disc ∆ = {|ζ| 1}, has its boundary

A ε ′ z 2 (∂∆) = A ε ′ z 2 {|ζ| = 1} contained in H 2-1
ε , the set where f is defined. Lowering dimensions by a unit, we draw discs as (green) segments and boundaries of discs as (green) bold points. Thus, we may compute the Cauchy integral

F (z 1 , z 2 ) := 1 2πi ∂∆ f A ε ′ z 2 (ζ) ζ -z 1 dζ.
Differentiating under the sum, the function F is seen to be holomorphic. In addition, for

|z 2 | < ε, it coincides with f , because the full closed disc A ε ′ z 2 ∆ is contained in H 2-1 ε
and thanks to Cauchy's formula. Clearly, the A ε ′ z 2 (∆) all together fill in the bidisc ∆ 2 . One may think that, as z 2 varies, discs are "pushed" gently by a virtual thumb.

Levi extension and the Levi-Hartogs figure.

Geometrically, the standard Hartogs figure is not best suited to perform holomorphic extension from a strongly (pseudo)concave boundary. For instance, in the proof of Theorem 2.7, we will encounter complements in C n of some closed balls whose radius decreases step by step, and more generally spherical shells whose thickness increases interiorly. Thus, we delineate an appropriate set up.

For r ∈ R with r > 1 and for δ ∈ R with 0 < δ < < 1, the sphere S 2n-1 r = {z ∈ C n : ||z|| = r} of radius r is the interior (and strongly concave) boundary component of the spherical shell domain , we could (for instance) restraint our considerations to some half-cylinder of diameter ≈ δ, but it will be better to shape a convenient half parallelepiped. Accordingly, for two small ε j > 0, j = 1, 2, we introduce a geometrically relevant Levi-Hartogs figure (right illustration, reverse orientation):

S r+δ r := r < ||z|| < r + δ = p∈S 2n-1 r B n (p, δ) ∩ {||z|| > r}. C n x n y n z ′ ε1 ε1 -(ε1) 2 ε1 ε2 LHε 1 ,ε 2 0 p S r+δ r S 2n-1 r B n
LH ε 1 ,ε 2 := max 1 i n-1 |z i | < ε 1 , |x n | < ε 1 , -ε 2 < y n < 0 ε 1 -(ε 1 ) 2 < max 1 i n-1 |z i | < ε 1 , |x n | < ε 1 |y n | < ε 2 .
To fill in this (bed-like) figure, we just compute the Cauchy integral on appropriate analytic discs (the (green) horizontal ones) whose boundaries remain in

LH ε 1 ,ε 2 . Lemma 3.4. O LH ε 1 ,ε 2 extends holomorphically to the full parallelepiped LH ε 1 ,ε 2 := max 1 i n-1 |z i | < ε 1 , |x n | < ε 1 , |y n | < ε 2 .
Next, we must reorient and scale LH ε 1 ,ε 2 in order to put it inside the shell. For every point p ∈ S 2n-1 r , there exists some complex unitarian affine map But we have to insure that Φ p LH ε 1 ,ε 2 as a whole (including the thin walls) lies outside B n r . Lemma 3.5. If ε 1 = c δ and ε 2 = c δ 2 with some appropriate 5

Φ p : z -→ p + Uz, with U ∈ SU(n, C), sending the origin 0 ∈ LH ε 1 ,ε 2 to p and T 0 LH ε 1 ,ε 2 to T p S 2n-1
positive constant c < 1, then Φ p LH ε 1 ,ε 2 is entirely contained in the shell S r+δ r . Furthermore, Φ p LH ε 1 ,ε 2 contains a rind of thickness c δ 2 r around some region R p ⊂ S 2n-1 r whose (2n -1)-dimensional area equals ≃ c δ 2n-1 . p Φp LHε 1 ,ε 2 p z ′ yn xn S r+δ r S r+δ r cδ c δ 2 r cδ 2n-2 Rp S 2n-1 r c δ 2 r c δ 2

, we mean

Rind R,

η := (1 + s)z : z ∈ R, |s| < η/r .
We require that |s| < η/r to insure that at every z ∈ R, the half-line (0z) + emanating from the origin intersects Rind R, η along a symmetric segment of length 2 η centered at z. In the diagram above, we draw (in green) only the lower part of the small region R p got in Lemma 3. The rigorous proof of the lemma (not developed here) involves elementary reasonings with geometric inequalities and a dry explicit control of the constants that does not matter for the sequel. The main argument uses the fact that S 2n-1 r detaches quadratically from T p S 2n-1 r , similarly as the parabola y = -1 r x 2 separates from the line {y = 0} in R 2

x,y . Since the area of S 2n-1 with the neighborhood V δ (M), we shall only get small subregions of S 2n-1 r . Hopefully, thanks to our local Levi-Hartogs figures, we may obtain a suitable semi-global extensional statement, valuable for proper subsets of the shell S r+δ r whose shape is arbitrary. The next statement, not available by means of large discs, will be used a great number of times in the sequel.

r equals 2 π n (n-1)! r 2n-1 = C r 2n-1 , by covering S 2n-1 r with such adjusted R p ⊂ Φ p LH ε 1 ,ε 2 of area c δ
Shell r+δ r R ∪ N := C n B n r ∩ V δ R ∪ N = p∈R∪N B n (p, δ) ∩ {||z|| > r}
N = ∂R N = ∂R R R S 2n-1 r S 2n-1 r Shell r+δ r R ∪ N Rind R, c δ 2 r -1
Proof. We must control uniqueness of holomorphic extension (monodromy) into rinds covered by successively attached Levi-Hartogs figures. Noticing c δ 2 r -1 < < δ, the considered rinds are much thinner than the piece of shell.

Lemma 3.8. If R ′ ⊂ R is an arbitrary open subset and if R p ′ ⊂ Φ p ′ LH ε 1 ,ε 2
is a small Levi-Hartogs region centered at an arbitrary point p ′ ∈ R, then the intersection

(3.9) Rind R p ′ , c δ 2 r -1 Shell r+δ r R ∪ N Rind R ′ , c δ 2 r -1 is connected.
Admitting the lemma for a while, we pick a finite number m C area(R)

δ 2n-1 of points p 1 , . . . , p m ∈ R ∪ N such that the associated local regions R p k contained in the filled Levi-Hartogs figures Φ p k LH ε 1 ,ε 2 provided by Lemma 3.5 do cover R ∪ N, namely R p 1 ∪ • • • ∪ R pm ⊃ R ∪ N.
Starting with R ′ := ∅ and p ′ := p 1 , unique holomorphic extension of O Shell r+δ r (R ∪ N) to Rind R p ′ , c δ 2 r -1 holds by means of Lemma 3.4, monodromy being assured thanks to the connectedness of the intersection (3.9). Reasoning by induction, fixing some k with 1 k m -1, setting R ′ := ∪ 1 j k R p j , p ′ := p k+1 and assuming that unique holomorphic extension is got from Shell

r+δ r R ∪ N into Shell r+δ r R ∪ N Rind R ′ , c δ 2 r -1 = Shell r+δ r R ∪ N 1 j k Rind R p j , c δ 2 r -1 ,
we add the Levi-Hartogs figure Φ p k+1 LH ε 1 ,ε 2 constructed in Lemma 3.5, and we get unique holomorphic extension to Rind R p k+1 , c δ 2 r -1 , monodromy being assured again thanks to the connectedness of the intersection (3.9). Since Rind R, c δ 2 r -1 ⊂ 1 k m Rind R p k , c δ 2 r -1 , the proposition is proved.

Proof of Lemma 3.8. To establish connectedness of the open set (3.9), picking two arbitrary points q 0 , q 1 in it, we must produce a curve joining q 0 to q 1 inside (3.9). The two radial segments of length 2 c δ 2 r -1 passing through q 0 and q 1 that are centered at two appropriate points of S 2n-1 r are by definition both entirely contained in Rind R p ′ , c δ 2 r -1 as well as in Rind R ′ , c δ 2 r -1 . Thus, moving radially, we may join inside (3.9) q 0 to a new point q ′ 0 and q 1 to a new point q ′ 1 , which both belong to the upper half-rind

(1 + s) z : z ∈ R p ′ , 0 < s < c δ 2 r -1 r .
Since this upper half-rind is connected and contained in Shell r+δ r R ∪ N , we may finally join inside (3.9) the point q ′ 0 to q ′ 1 .

In the sequel, in order to avoids several gaps and traps, we will put emphasis on rigourously checking univalence of holomorphic extensions. §4. FILLING DOMAINS OUTSIDE BALLS OF DECREASING RADIUS 4.1. Global Levi-Hartogs filling from the farthest point. We can now launch the proof of Theorem 2.7. The δ 1 is first chosen so small that V δ (M) is a true tubular neighborhood of M for every δ with 0 < δ δ 1 . Shrinking even more δ 1 , in balls of radius δ 1 centered at its points, the hypersurface M is well approximated by its tangent planes.

The farthest point of Ω M from the origin is unique and it coincides with p κ since by assumption p κ is the single critical point of r(z) M with || p κ || = max 1 λ κ || p λ ||. By assumption also, the Hessian matrix of r(z) M is nondegenerate at p κ ; this also follows automatically from the inclusion Ω M ⊂ B n rκ , which constrains strong convexity of M at p κ . Consequently, according to the Morse lemma ( [START_REF] Milnor | Morse Theory, Based on lectures notes[END_REF], [START_REF] Hirsch | Differential topology[END_REF], Ch. 6), there exist local coordinates (θ 1 , . . . , θ 2n-1 ) on M centered at p κ such that the intersection M ∩ S 2n-1 r is given by the equation

-θ 2 1 -• • • -θ 2 2n-1 = r -r κ , for all r close to r κ . Thus M ∩ S 2n-1
r is empty for r > r κ ; it reduces to { p κ } for r = r κ ; and it is diffeomorphic to a (2n -2)-sphere for r < r κ close to r κ .

Similarly, the nearest point of Ω M from the origin is unique and it coincides with p 1 ; notice that hence κ 2. Also, the second farthest critical point p κ-1 lies at a distance r κ-1 < r κ from 0. If necessary, we shrink δ 1 to insure (4.2)

δ 1 < < min 1 λ κ-1 r λ+1 -r λ .
Next, for every radius r with r κ-1 < r < r κ , we introduce the cut out domain Since there are no critical points of r(z) M in the interval r κ-1 , r κ , Morse theory shows that M >r is a deformed spherical cap diffeomorphic to R 2n-1 for every r with r κ-1 < r < r κ . Also, Ω >r is then a piece of deformed ball diffeomorphic to R 2n .

Ω >r := Ω M ∩ ||z|| > r
The boundary in C n of Ω >r

∂Ω >r = M >r ∪ R r ∪ N r consists of M >r together with the open subregion R r := Ω M ∩ ||z|| = r of S 2n-1 r
which is diffeomorphic to R 2n-1 and has boundary N r := M ∩ ||z|| = r diffeomorphic to the unit (2n -2)-sphere. Thus, the global geometry of Ω >r is understood.

We can also cut out V δ (M), getting V δ (M) >r . The central figure shows that when r > r κ-1 is very close to r κ-1 , a parasitic connected component W >r of V δ (M) >r might appear near p κ-1 . After filling Ω >r progressively by means of Levi-Hartogs figures (see below), because Ω >r ∩ V δ (M) >r is not connected in such a situation, no unique holomorphic extension can be assured, and in fact, multivalence might well occur.

A trick to erase such parasitic components W >r is to consider instead the open set

V δ M >r >r = V δ M >r ∩ ||z|| > r ,
putting a double " >r ". It is drawn in the right figure and it is always diffeomorphic to M >r × (-δ, δ).

From pieces of shells as in Proposition 3.7 which embrace spheres of varying radius r, holomorphic extension holds to (symmetric) rinds whose thickness c δ r -1 also varies. To simplify, we introduce the smallest appearing thickness (4.3) η := min

r 1 r rκ c δ r -1 = c δ r κ -1 ,
and we observe that it follows trivially from Proposition 3.7 (just by shrinking and by restricting) that holomorphic extension holds to some rind around R of arbitrary smaller thickness η ′ > 0 with 0 < η ′ η. In the sequel, our rinds shall most often have the uniform thickness η, and sometimes also, a smaller one η ′ . Shrinking the constant c of η in (4.3), we insure η < < δ 1 .

Summarizing, we list and we compare the quantities introduced so far:

(4.4)                  0 < δ δ 1 neighborhood V δ (M ) 2 r( p 1 ) < • • • < r( p κ ) 5 + diam Ω M Morse radii δ δ 1 < < min 1 λ κ-1 r λ+1 -r λ smallness of V δ (M ) η := c δ 2 r -1 κ uniform useful rind thickness η < < δ
thickness of extensional rinds is tiny Proposition 4.5. For every cutting radius r with r κ-1 < r < r κ arbitrarily close to r κ-1 , holomorphic functions in the open set Proof. We fix such a radius r with r κ-1 < r < r κ . Putting a single Levi-Hartogs figure at p κ as in Proposition 3.7, we get unique holomorphic extension to Ω > rκ-η . Since η < < δ, we have r κη > r κ-1 . If the radius r κη is already < r, we just shrink to η ′ := r κr < η the thickness of our single rind, getting unique holomorphic extension to Ω > rκ-η ′ = Ω >r . Performing induction on an auxiliary integer k 1, we suppose that, by descending from r κ to a lower radius r ′ := r κkη assumed to be still r, holomorphic functions in V δ M >r >r extend holomorphically and uniquely (remind Definition 2.5) to Ω >r ′ . Lemma 4.6. For every radius r ′ with r κ-1 < r < r ′ < r κ ,

V δ M >r >r = V δ M >r ∩ ||z|| > r
(4.7) Shell r ′ +δ r ′ R r ′ ∪ N r ′ is contained in Ω >r ′ V δ M >r >r .
Fig. 9: A shell contained in the cap-shaped domain and the associated rind

Shell r ′ +δ r ′ R r ′ ∪ N r ′ ΩM r ′ r r ′ r M M Ω >r ′ -η pκ pκ Ω >r ′ Rind -R r ′ , η
Proof. Picking an arbitrary point p ∈ R r ′ ∪ N r ′ , we must verify that

B n (p, δ) ∩ {||z|| > r ′ }
is contained in the right hand side of (4.7).

If p ∈ N r ′ ⊂ M, whence p ∈ M >r , we get simply what we want:

B n (p, δ) ∩ {||z|| > r ′ } ⊂ V δ M >r ∩ {||z|| > r ′ } ⊂ V δ M >r ∩ {||z|| > r} = V δ M >r >r .
If p ∈ R r ′ N r ′ , whence p ∈ Ω M , reasoning by contradiction, we assume that there exists a point q ∈ B n (p, δ) ∩ {||z|| > r ′ } in the cut out ball which does not belong to the right hand side of (4.7). Since

Ω >r ′ = Ω M ∩ {||z|| > r ′ }, we have q ∈ Ω M . Reminding R r ′ ⊂ S 2n-1 r ′
, the tangent plane T p S 2n-1

r ′ = T p R r ′ divides C n in two closed half-spaces, T + p S 2n-1 r ′ exterior to B n r ′ and the opposite one T - p S 2n-1 r ′
. We distinguish two (nonexclusive) cases. p q p q q p p Fig. 10: Checking that the shell is contained in the cut out domain

M >r ′ r ′ R r ′ r ′ N r ′ M >r ′ R r ′ Ω >r ′ Ω >r ′ Firstly, suppose that the half-line (pq) + is contained in T + p S 2n-1 r ′
, as in the left figure. Since p ∈ Ω M and q ∈ Ω M , there exists at least one point p of the open segment (p, q) which belongs to M, hence p ∈ M >r . Then ||q -p|| < ||q -p|| < δ, whence q ∈ B n ( p, δ) ∩ {||z|| > r} and we deduce that q ∈ V δ M >r >r belongs to the right hand side of (4.7), contradiction.

Secondly, suppose that the half-line (pq

) + is contained in T - p S 2n-1 r ′
, as in the right figure. Let q ∈ (p, q) be the middle point. In the plane passing through 0, p and q, consider a circle passing through p and q and centered at some point close to 0 in the open segment (0, q). It has radius < r ′ close to r ′ . The open arc of circle between p and q is fully contained in {||z|| > r ′ }.

Since p ∈ Ω M and q ∈ Ω M , there exists at least one point p of the open arc of circle between p and q which belongs to M, hence p ∈ M >r . But then (p, q) is the hypothenuse of the triangle pq p (remind r ′ > 1 and ||q -p|| < δ < < 1), whence ||q -p|| < ||q -p|| < δ, hence again as in the first case, we deduce that q ∈ V δ M >r >r , contradiction.

If the slightly smaller radius

r ′′ := r ′ -η = r κ -(k + 1)η
is already < r, we will shrink to η ′ := r κrkη < η the thickness of the final extensional rind. Otherwise, in the generic case, r κ -(k + 1)η is still > r. The final (exceptional) case being formally similar, we continue the proof with r ′ = r κkη and r ′′ = r ′η, assuming that r ′′ r.

Setting r ′ := r κkη in the auxiliary Lemma 4.6, functions holomorphic in Ω >r ′ ∪ V δ M >r >r restrict to Shell r ′ +δ r ′ R r ′ ∪ N r ′ and then, thanks to Proposition 3.7, these restricted functions extend holomorphically to Rind R r ′ , η .

(4.9) Rind R r ′ , η Ω >r ′ ∪ V δ M >r >r .
Furthermore, the union of the same two open sets contains

(4.10) Ω >r ′ -η ∪ V δ M >r >r .
Thus we get unique holomorphic extension to (4.10) and finally, by induction on k and taking account of the final step where η should be shrunk appropriately, we get unique holomorphic extension to Ω >r ∪ V δ M >r >r .

The number of utilized Levi-Hartogs figures is majorated by the product of the number of needed rinds ∼ rκ-r η times the maximal area of R r ′ , which we roughly majorate by the area C ( r κ ) 2n-1 of the biggest sphere S 2n-1 rκ , everything being divided by the area c δ 2n-1 covered by a small Levi-Hartogs figure. This yields the finite number claimed in Proposition 4.5, achieving its proof.

Proof of Lemma 4.8. [May be skipped in a first reading] To establish connectedness, we decompose the rind as

Rind + := (1 + s)z : z ∈ R r ′ , 0 < s < η/r ′ Rind 0 := R r ′ , Rind -:= (1 -s)z : z ∈ R r ′ , 0 < s < η/r ′ , so that Rind = Rind -∪ Rind 0 ∪ Rind + , without writing the common argument R r ′ , η .
Obviously, the upper Rind + is diffeomorphic to R r ′ × (0, η) ≃ R 2n-1 × (0, η), hence is connected. We claim that, moreover, the full Rind + is contained in Ω >r ′ ∪ V δ M >r >r , whence (4.11)

Rind + = Rind + Ω >r ′ ∪ V δ M >r >r .
Indeed, let q ′ ∈ Rind + , hence of the form q ′ = (1+s)p ′ for some p ′ ∈ Rind 0 = R r ′ and some s with

0 < s < η/r ′ . If the half-open-closed segment (p ′ , q ′ ] is contained in Ω M , hence in Ω >r ′ = Ω M ∩ ||z|| > r ′ , we get for free q ′ ∈ Ω >r ′ .
If on the contrary, (p ′ , q ′ ] is not contained in Ω M , then there exists a point q ′ ∈ (p ′ , q ′ ] with q ′ ∈ M = ∂Ω M , whence q ′ ∈ M >r ′ ⊂ M >r (remind r ′ -η r). The ball B n ( q ′ , δ) then contains q ′ , because ||q ′q ′ || < ||q ′p ′ || η < < δ. This shows q ′ ∈ V δ M >r >r , achieving the claim.

Thus, the (upper) subpart (4.11) of the intersection (4.9) is already connected.

To conclude the proof of connectedness, it suffices to show that every point p ′ of the remaining part (4.12)

Rind 0 ∪ Rind - Ω >r ′ ∪ V δ M >r >r can be joined, by means of some appropriate continuous curve running inside the intersection (4.9), to some point q ′ of the connected upper subpart (4.11). Thus, let p ′ in (4.12) be arbitrary. If p ′ ∈ Rind 0 ∩ Ω >r ′ ∪ V δ M >r >r , it suffices to join radially p ′ to q ′ = (1 + s ε )p ′ , for some s ε with 0 < s ε < < η. Indeed, such a q ′ then belongs to Rind + ∩ Ω >r ′ ∪ V δ M >r >r .

If p ′ ∈ Rind -∩ Ω >r ′ ∪ V δ M >r >r , then necessarily p ′ ∈ V δ M >r >r , because by definition:

Rind -R r ′ , η ∩ Ω >r ′ = ∅.
So there is a point q ∈ M >r with p ′ ∈ B n (q, δ).

r ′ Rind R r ′ , η q V δ M>r >r M>r p ′ r ′ p ′ Rind R r ′ , η M>r V δ M>r >r B n (q, δ) B n (q, δ) q q ′ t p ′ t q ′ q ′′ q ′
We then distinguish two exclusive cases: either r(q) r ′ or r(q) < r ′ . Firstly, assume r(q) r ′ (left diagram). If 0, p ′ and q are aligned, we simply join p ′ to the point q ′ := (1 + s ε ) r ′ r(p ′ ) p ′ which belongs to Rind + . The segment [p ′ , q ′ ] is then entirely contained in Rind ∩ B n (q, δ) >r , hence in (4.9).

Otherwise, in the unique plane passing through 0, p ′ and q, consider the point q ′′ := r(p ′ ) r(q) q, satisfying r(q ′′ ) = r(p ′ ) and belonging to (0, q). Since q ′′ is the orthogonal projection of q onto B n (0, r(p ′ )), we get ||qq ′′ || < ||qp ′ || < δ, whence q ′′ ∈ B n (q, δ). The circle of radius r(p ′ ) centered at 0 joins p ′ to q ′′ by means of a small arc which is entirely contained in B n (q, δ). Denote by γ : [0, 1] → B n (q, δ) a parametrization of this arc of circle, with γ(0) = p ′ and γ(1) = q ′′ . If γ[0, 1] is entirely contained in Rind -, we conclude by joining q ′′ radially to the point q ′ := (1 + s ε ) r ′ r(q ′′ ) q ′′ . If γ[0, 1] is not contained in Rind, let t 1 ∈ (0, 1) satisfying γ[0, t 1 ) ⊂ Rind - but γ(t 1 ) ∈ Rind -. Then γ(t 1 ) belongs to ∂Rind -and since r(γ(t 1 )) = r(p ′ ) still satisfies r ′η < r(p ′ ) < r ′ , necessarily γ(t 1 ) belongs "vertical part" of ∂Rind -, namely to the strip (1s)z : z ∈ N r ′ , 0 s η/r ′ . Hence the point q ′′′ := r ′ r(γ(t 1 )) γ(t 1 ) belongs to N r ′ . We now modify γ by constructing a curve which remains entirely inside B n (q ′′′ , δ) >r ⊂ V δ M >r >r as follows: choose t 2 < t 1 very close to t 1 , join p ′ to γ(t 2 ) ∈ Rind -through γ and then γ(t 2 ) radially to the point q ′ := (1 + s ε ) r ′ r(γ(t 2 )) γ(t 2 ) ∈ Rind + . The resulting curve is entirely contained in (4.9). In conclusion, we have joined p ′ to a suitable point q ′ , as announced.

Secondly, assume that r(q) < r ′ . Consider the normalized gradient vector field ∇r M ||∇r M || , defined and nowhere singular on M ∩ r κ-1 < ||z|| < r κ , hence on M >r { p κ }. For t ∈ [0, 2 η], denote by t → q t the integral curve of ∇r M ||∇r M || passing through q, satisfying q 0 = q, q t ∈ M and r(q t ) = r(q) + t. Together with its center q, the ball is translated as B n (q t , δ). Accordingly, the point p ′ is moved, yielding a curve p ′ t such that p ′ t occupies a fixed position with respect to the moving ball. Explicitly: p ′ t = p ′ + q ′ tq. Thanks to r ′ > 1 and δ < < 1, one may check 6 that dr(p ′ t ) dt 1c r ′ ,δ , for some small positive constant c r ′ ,δ <1. Thus, as t increases, the point p ′ t moves away from 0 at speed almost equal to 1. Since r ′η < r(p ′ 0 ) < r ′ , we deduce that for t = 2 η, we have r(p ′ 2η ) > r ′ , namely p ′ 2η has escaped from Rind -. Consequently, there exists t 1 ∈ (0, 2η) with p ′ t ∈ Rind -for 0 t < t 1 such that p ′ t 1 ∈ ∂Rind -. The boundary of Rind -has three parts: the top R r ′ , the bottom (1η/r ′ )z : z ∈ R r ′ and the (closed) strip (1s)z : z ∈ N r ′ , 0 s η/r ′ . The limit point p ′ t 1 cannot belong to the bottom, since r(p ′ t 1 ) > r(p ′ 0 ) > r ′η. Since by construction p ′ t ∈ B n (q t , δ) with q t ∈ M >r , we observe that p ′ t ∈ V δ M >r >r for every t ∈ [0, 2 η]. Consequently:

p ′ t ∈ Rind -V δ M >r >r , ∀ t ∈ [0, t 1 ).
Assuming that p ′ t 1 ∈ ∂Rind -belongs to the top R r ′ = Rind 0 , we may join p ′ t 1 radially to q ′ := (1 + s ε )p ′ t 1 . In this way, p ′ is joined, by means of a continuous curve running in the intersection (4.9), to the point q ′ = (1 + s ε )p ′ t 1 belonging to the connected upper subpart (4.11).

Finally, assume that p ′ t 1 ∈ ∂Rind -belongs to the strip (1s)z : z ∈ N r ′ , 0 s η/r ′ . The point q ′′ := r ′ r(p ′ t 1

) p ′ t 1 belongs to N r ′ ⊂ M >r , and we will construct a small curve running entirely inside B n (q ′′ , δ) >r ⊂ V δ M >r >r . Choose t 2 ∈ (0, t 1 ) very close to t 1 , join p ′ to p ′ t 2 ∈ Rind -as above (but do not go up to p ′ t 1 ) and then join p ′ t 2 radially to the point q

′ := (1 + s ε ) r ′ r(p ′ t 2
) p ′ t 2 , which belongs to Rind + . The small radial segment from p ′ t 2 to q ′ is entirely contained in B n (q ′′ , δ) and in the full Rind. In conclusion, p ′ is joined, by means of a continuous curve running in the intersection (4.9), to this point

q ′ = (1 + s ε ) r ′ r(p ′ t 2
) p ′ t 2 which belongs to the connected upper subpart (4.11). The proof of the connectedness of the intersection (4.9) is complete.

We now show that the union, instead of the intersection in (4.9), contains (4.10). [START_REF] Chirka | Removable singularities in the boundary[END_REF] If the spheres S 2n-1 r for r close to r ′ would be hyperplanes -they almost are in comparison to B n (q t , δ) -we would have exactly r(p ′ t ) = r(p ′ ) + t, whence

dr(p ′ t ) dt = 1. Let p ′ ∈ Ω >r ′ -η \Ω >r ′ , whence r ′ -η < ||p ′ ||
r ′ . The radial half line t p ′ : 0 < t < ∞ emanating from the origin and passing through p ′ meets S 2n-1

r ′ at the point q ′ = r ′ ||p ′ || p ′ . If the closed segment [p ′ , q ′ ] is contained in Ω >r ′ -η , then q ′ ∈ Ω M . Since ||q ′ || = r ′ and since R r ′ = Ω M ∩ ||z|| = r ′ , we get q ′ ∈ R r ′ , whence p ′ ∈ Rind R r ′ , η .
If on the contrary, the closed segment [p ′ , q ′ ] is not contained in Ω >r ′ -η , then there exists q ′ ∈ (p ′ , q ′ ] with q ′ ∈ M = ∂Ω M , whence q ′ ∈ M >r ′ -η ⊂ M >r . Since η < < δ, we deduce p ′ ∈ B n ( q ′ , δ) and finally p ′ ∈ V δ M >r >r .

The proofs of Lemma 4.8 and hence also of Proposition 4.5 are complete. §5. CREATING DOMAINS, MERGING AND SUPPRESSING CONNECTED COMPONENTS

Topological stability and global extensional geometry between regular

values of r M . In the preceding Section 4, for r with r κ-1 < r < r κ , we described the simple shape of the cut out domain Ω >r = Ω M ∩ {||z|| > r}, just diffeomorphic to a piece of ball. Decreasing the radius under r κ-1 , the topological picture becomes more complex. At least for radii comprised between two singular values of r(z) M , Morse theory assures geometrical control together with constancy properties.

Lemma 5.2. Fix a radius r satisfying r λ < r < r λ+1 for some λ with 1 λ κ -1, hence noncritical for the distance function r(z)| M . Then:

(a) T z M + T z S 2n-1 r = T z C n at every point z ∈ M ∩ S 2n-1 r ; (b) the intersection M ∩ S 2n-1 r is a C ∞ compact hypersurface N r ⊂ S 2n-1 r of codimension 2 in C n ,

without boundary and having finitely many connected components;

(c) N r ′′ is diffeomorphic to N r ′ , whenever r λ < r ′′ < r ′ < r λ+1 ; (d) M >r = M ∩ {||z|| > r} has finitely many connected components M c >r , with 1 c c λ , for some c λ < ∞ which is independent of r; (e) M c >r ′′ is diffeomorphic to M c >r ′ , whenever r λ < r ′′ < r ′ < r λ+1 , for all c with 1 c c λ ;

(f) M ∩ {r ′′ < ||z|| < r ′ } is diffeomorphic to N r ′ × (r ′′ , r ′ ), hence also to N r ′′ × (r ′′ , r ′ ), whenever r λ < r ′′ < r ′ < r λ+1 ;
Proof. We summarize the known arguments of proof (cf. [START_REF] Milnor | Morse Theory, Based on lectures notes[END_REF] and [START_REF] Hirsch | Differential topology[END_REF], Ch. 6). Equivalently, (a) says that dr : field which is obviously orthogonal to spheres and consider the orthogonal projection X D of D| M on T M, a C ∞ vector field on M. We want to scale the gradient as V r,M := λ • ∇(r| M ) so that its radial component is identically equal to one, namely, so that V r,M , D ≡ 1, which gives the equation:

T z M → T r(z) R
1 = λ ∇(r| M ), D = λ ∇(r| M ), X D = λ ∇(r| M ), X D M = λ d r| M (X D ).
To simply set λ :=

1 d(r| M )(X D )
, we must establish that X D cannot belong to Ker d r| M at any point z ∈ M ∩ r λ < ||z|| < r λ+1 of a noncritical shell.

We check this. At such a point z, D(z) is not orthogonal to T z M (otherwise ||z|| , then it would be orthogonal to D(z), and in the plane Π z , we would have both D(z) and the hypothenuse D -X D (z) orthogonal to X D (z), which is impossible.

T z M would coincide with T z S 2n-1 ||z|| ), whence its orthogonal projection X D (z) is = 0. By definition, (D -X D )(z) is orthogonal to T z M ∋ X D (z), hence it is orthogonal to X D (z)
Thus, in spherical coordinates (r, ϑ 1 , . . . , ϑ 2n-1 ) restricted to a noncritical shell, the r-component of the C ∞ scaled gradient vector field V r,M :=

∇(r| M ) (∇(r| M ), D)
is ≡ 1. We deduce that the flow (wherever defined) z s := exp(s V r,M (z) simply increases the norm as

||z s || = ||z|| + s, whence exp (r ′ -r ′′ )V r,M (•) in- duces a diffeomorphism from N r ′′ onto N r ′ : this yields (c). Also, (z ′′ , s) -→ exp (r ′′ + s)V r,M (z ′′ ) gives the diffeomorphism of N r ′′ × (r ′ -r ′′ ) onto the strip M ∩ r ′′ < ||z|| < r ′ , which is (f).
Next, the compact manifold with boundary M >r ∪ N r surely has finitely many connected components, whose number is constant for all r λ < r < r λ+1 , because when r increases or decreases, the connected components of the slices N r do slide smoothly in S 2n-1 r without encountering each other: this is (d). Finally, (e) follows from (f) and the trivial fact that the two segments (r ′′ , r 0 ) and (r ′ , r 0 ) are diffeomorphic, whenever r λ < r ′′ < r ′ < r 0 < r λ+1 .

We can now state the very main technical proposition of this paper. We point out that in (i) and (ii), neither Ω c r nor R c r need be contained in our original domain Ω M (as it was the case in Section 4 for r κ-1 < r < r κ ): this is why we introduced a widetilde notation. We refer to the middle Figure 1 for an illustration. Similarly, neither Ω c r nor R c r need be contained in C n Ω M : they both may intersect Ω M and C n Ω M . Also, the number of connected components of N c r is that of R c r and may be >, as illustrated below.

∀ f ∈ O V δ (M) ∃ F ∈ O Ω M ∪ V δ (M) s.t. F V δ (M ) = f.
Proof. In the main Proposition 5.3, we choose r = r 1 + ε (where ε > 0 satisfies ε < < δ) very close to the last, smallest singular radius. Then M >r has a single connected component, M >r itself, and it simply bounds Ω M >r . The remainder part of M, namely M ∩ ||z|| r 1 + ε is diffeomorphic to a very small closed (2n -1)-dimensional spherical cap and is entirely contained in V δ (M).

Fix an arbitrary function f ∈ O V δ (M) and restrict it to V δ M >r >r . Thanks to the proposition, f extend holomorphically and uniquely to Ω M >r by means of a finite number of Levi-Hartogs figures. Since

V δ (M) V δ M >r >r ∪ Ω M >r
is easily seen to be connected, we get a globally defined extended function which is holomorphic in

V δ (M) V δ M >r >r ∪ Ω M >r = V δ (M) ∪ Ω M .
This completes the proof.

Proof of Proposition 5.3. In (i), let us check the uniqueness of a relatively compact Ω c >r . Since M c >r inherits an orientation from M, the complement ||z|| > r M c >r has at most 2 connected components. As M ⋐ C n is bounded, at least one component contains the points at infinity, hence there can remain at most one component of ||z|| > r M c >r that is relatively compact in C n . If r satisfies r κ-1 < r < r κ , Proposition 4.5 already completes the proof.

Assume therefore that r satisfies r µ < r < r µ+1 , for some µ ∈ N with 1 µ κ -1. For every λ with 2 λ κ -1, it will be convenient to flank each singular radius r λ by the following two very close nonsingular radii

(5.5) r - λ := r λ -η/2 and r + λ := r λ + η/2 ,
with η being the same uniform thickness of extensional rinds as before. We fix once for all an arbitrary function f holomorphic in V δ M >r >r . Letting λ be arbitrary with µ λ κ -1, the logic of the proof shows up two topologically distinct phenomena that we overview.

A: Filling domains through regular radii intervals. Assume that at the regular radius r - λ+1 = r λ+1 -η 2 , all domains Ω c > r - (II) Secondly, assume k λ = 1. This is the most delicate case, because in a small neighborhood of p λ , the cut out hypersurface M > r + λ has exactly 2 connected components, so that two different enclosed domains

Ω c 1 > r + λ and Ω c 2 > r + λ
can meet here; it may also occur that the two parts near p λ belong to the same domain, i.e. that c 2 = c 1 . While descending down to r - λ , we must analyze the way how the two (maybe the single) component(s) merge. Three subcases will be distinguished, one of which showing a crucial trick of subtracting one growing component from a larger one which also grows (right Figure 1).

(III) Thirdly, assume that 2 k λ 2n -2. In all these cases, locally in a neighborhood of p λ , the cut out hypersurface M > r + λ has exactly 1 connected component and the way how the corresponding single enclosed domain Ω c > r + λ grows will be topologically constant.

Reasoning by induction on λ and applying the filling processes A and B, we then descend progressively inside deeper spherical shells, checking all properties of Proposition 5.3. When approaching the bottom radius r of Proposition 5.3, it will suffice to shortcut A or B appropriately in order to complete the proof.

Filling domains through regular radii intervals.

Recall that r µ < r < r µ+1 , let λ with µ λ κ-1 and consider the regular radius interval r + λ , r - λ+1 . We suppose first that r r + λ , so that we may descend inside the whole spherical shell r + λ < ||z|| r - λ+1 . Afterwards, we explain how we stop in the case where λ = µ and r + µ < r < r - µ+1 . By descending induction on λ through A and B, we may assume that at the superlevel set

(•) > r - λ+1 , the domains Ω c > r - λ+1 enclosed by M c > r - λ+1 for 1 c c λ have been constructed and that each restriction f c r - λ+1 of f ∈ O V δ M >r >r to V δ M c > r - λ+1 > r - λ+1
extends holomorphically and uniquely to the domain

(5.7) Ω c > r - λ+1 V δ M c > r - λ+1 > r - λ+1 .
For every radius r ′ with r + λ r ′ < r - λ+1 , the cut out hypersurface

M >r ′ = 1 c c λ M c >r ′ has the same number of connected components, each M c >r ′ is diffeomorphic to M c > r - λ+1 and the difference M c >r ′ M c > r - λ+1 is diffeomorphic to N c r - λ+1 × r ′ , r - λ+1 . Furthermore, each prolongation Ω c >r ′ of Ω c > r -
V δ M c > r + λ r + λ by adding legs to V δ M c > r - λ+1 r - λ+1 p λ+1 p λ r + λ r - λ+1 V δ M c > r - λ+1 > r - λ+1 V δ M c > r + λ > r + λ r ′
Since f was defined in V δ M >r >r and since r r + λ , we claim that each restriction f c r - λ+1 may be extended holomorphically and uniquely to (5.8)

Ω c > r - λ+1 V δ M c > r + λ > r + λ .
Indeed, to the original domain of definition (5.7) of f c r - λ+1 which was contained in ||z|| > r - λ+1 , we add in the enlarged domain (5.8) a finite number j λ,c of tubular domains around the connected components of M c >r ′ M c > r -

λ+1

. Because δ was chosen so small that V δ (M) is a small tubular neighborhood of M, and because f ∈ O V δ M >r >r is uniquely defined, we get a unique extension, still denoted by f c r - λ+1 , to (5.8). We can now apply the same reasoning as in Proposition 4.5, which consists of progressive holomorphic extension by means of thin rinds. Reproducing the proof of Lemma 4.6 (with changes of notation only), we get for every radius r ′ with r + λ < r ′ r - λ+1 that (5.9)

Shell r ′ +δ r ′ R c r ′ ∪ N c r ′ is contained in Ω c >r ′ V δ M c > r + λ > r + λ .
Similarly, reproducing the proof of Lemma 4.8 yields the connectedness of

Rind R c r ′ , η Ω c >r ′ ∪ V δ M c > r + λ > r + λ ,
and furthermore, this yields that the union, instead of the intersection, contains

Ω c >r ′ -η V δ M c > r + λ > r + λ , whenever r ′ -η is still r + λ (otherwise,
V δ M c > r + λ > r + λ Ω c > r + λ .
Finally, if r satisfies r + µ < r < r - µ+1 , descending from (•) > r - µ+1 with λ = µ as above, we just stop the construction of rinds to (•) >r by shrinking appropriately the thickness of the last extensional rind.

The property (iii) that enclosed domains Ω c >r are either disjoint or one is contained in the other remains stable as r decreases through the whole nonsingular interval r λ , r λ+1 , because their (moving) boundaries always remain disjoint, so that property (iv) is also simultaneously transmitted to lower regular radii. This completes A.

Localizing (pseudo)cubes at Morse points.

We now study B. Recall that r µ < r < r µ+1 , let λ with µ λ κ -1 and suppose that r r - λ , so that starting from (•) > r + λ , we may (and we must) continue the Hartogs-Levi filling inside the whole thin spherical shell r - λ < ||z|| r + λ . Similarly as above, the way how we should stop the process in the case where λ = µ and r µ < r < r + µ is obvious.

By descending induction on λ through A and B, we may assume that at r + λ , the domains

Ω c > r + λ enclosed by M c > r + λ for 1 c c λ have been constructed
and that each restriction

f c r + λ of f ∈ O V δ M c >r >r to V δ M c > r + λ > r +
λ extends holomorphically to the domain (5.10) of the previous paragraph.

By an elementary analysis of the Morse normalizing quadric, we will see that in some small (pseudo)cube centered at p λ , there passes in most cases only one component M c > r + λ , while in a single exceptional case, there can pass two (at most) different connected components M c 1 > r + λ and M c 2 > r + λ . We will consider only this single (or these two) component(s), because the other components do pass regularly and without topological change accross p λ , hence are filled in by Levi-Hartogs figures exactly as in A.

Shrinking the δ 1 of Theorem 2.7 if necessary (remind 0 < δ δ 1 ), we may assume that the Morse normalizing coordinates v, x 1 , . . . , x k λ , y 1 , . . . , y 2n-1-k λ near p λ are defined in the ball B n ( p λ , δ 1 ) and that the map

z -→ v(z), x(z), y(z) , B n ( p λ , δ 1 ) -→ R 2n
is close in C 1 norm to its differential at p λ , so that it is almost not distorting. Then δ 1 shall not be shrunk anymore.

Because in the estimates of the (finite) number of Levi-Hartogs figures, η only appears as a denominator in a factor r ′ -r ′′ η (cf. Proposition 4.5), it is allowed to work with extensional rinds of smaller universal positive thickness, at the cost of spending a number of pushed analytic discs that is greater, of course, but still finite. If necessary, we shrink η > 0 to insure that η 1/2 < < δ. Then η will not be shrunk anymore.

Thanks to these preliminaries, we may define a convenient (pseudo)cube centered at p λ by (5.12)

C η := z ∈ B n ( p λ , δ 1 ) : |v(z)| < η, ||x(z)|| < 2 η 1/2 , ||y(z)|| < 2 η 1/2 .
It then follows that C η is properly contained in V δ (M) and is relatively small. Reminding that v(z) = r(z)r( p λ ), the radial thickness of C η is equal to 2η, twice the difference r + λr - λ = η. We draw a diagram assuming k λ = 2n -1 (see only the left one).

V δ (M ) Fig. 14: The radial (pseudo)cube In a while, we will see that there is a salient topological difference between the two remaining (less obvious) cases 2 k 2n -2 and k = 1, the exceptional one. Before pursuing, we conclude the proof of B in case p λ is a local maximum or minimum.

C η centered at p λ v p λ M x x Cη v V δ (M ) x r + λ r - λ Cη r λ M c > r λ + M c > r λ + Ω c > r λ + 4η 1/2 2η 1/2 Rind - p λ Rind - η/2 η/2 η/2 η/2
We assume k λ = 2n -1, the case k λ = 0 being already considered (essentially completely) in Section 4. Observe that

M > r + λ ∩ C η is diffeomorphic to S 2n-2 × (c
r λ Rind - r + λ p λ r - λ Ω c > r + λ V δ M c > r + λ > r + λ M c > r + λ R c,2 r + λ R c,1 r + λ Rind -
We do the numbering so that C η encloses the first (small) N c,1 r + λ , which is diffeomorphic to a small (2n -2)-dimensional sphere. Also, we number so that the boundary of R c,1

r + λ in S 2n-1 r + λ contains N c,1 r + λ , whence R c,1 r + λ meets C η . We do not draw C η .
Observe that, by means of extensional rinds that are symmetric around the other components R

lying above v = η/2 but below v = x 2 1 +• • •+x 2 2n-1 ; the dimension of S 2n-1 r + λ being 3, the region R c,1 r + λ ∩ C η is connected, a fact that a one- dimensional diagram cannot show adequately; then Ω c > r + λ ∩ C η is also connected.
In case (a), near p λ , a piece of Ω c > r + λ ends up while descending to the lower super-level set (•) > r - λ . We do not use any extensional rind there, we just observe that unique holomorphic extension is got for free in

V δ M c > r - λ > r - λ ∩ C η ,
since this domain is fully contained in V δ M >r >r .

7

Sets written "{•}" here are understood to be subsets of C η .

In case (b), we apply Hartogs Levi extension to Rind R c,1 r + λ , η and we get unique holomorphic extension from (5.10) to The proof of B in case k λ = 2n -1 is complete. The case k λ = 0 is similar: two subcases (a') -reverse (a) -and (b') -reverse (b) -then appear; subcase (a') exhibits the birth of a new component (blue left Figure 15), as already fully studied in Section 4 while subcase (b') (green left Figure 15) shows that an external component descends regularly as do clouds around a hill.

V δ M c > r - λ > r - λ Rind R c,1 r + λ , η .

The regular cases 2

k λ 2n -2. Let k with 2 k 2n -2 and consider the quadric Q k of (5.14). We claim that Q k ∩ v > ε has exactly one connected component for every ε > 0. Indeed, Q k ∩ v > ε can be represented as

y 1 ,...,y 2n-k-1 ε ′ >ε x 2 1 + • • • + x 2 k = ε ′ + y 2 1 + • • • + y 2 2n-1-k .
Since ε ′ is always positive, we hence have a smoothly parameterized family of (k -1)-dimensional spheres that are all connected. Consequently, the union is also connected, as claimed.

To view the topology more adequately, in the case n = 2, we draw a short movie consisting of the 3-dimensional slices

v = ε ′ ∩ R 2n Q k , where ε ′ = 2 3 η, 1 2 η, 0, -1 2 η. To conceptualize (in case n = 2) the super-level sets v > ε ∩ R 2n Q k = ε ′ >ε v = ε ′ ∩ R 2n Q k ,
it suffices to pile up intuitively the images of the corresponding movie.

V δ M c > r - λ > r - λ Rind R c,1 r + λ , η .
The intersection of (5.

p λ r - λ r + λ R c 1 ,1 r + λ R c 2 ,1 r + λ R * r - λ {v = ε}
For every ε satisfying 0 < ε < η, there are two connected components M - >ε and M + >ε of M > r λ +ε ∩ C η , namely the two upper tips of the saddle, defined in equations by

M ± >ε := v = x 2 -y 2 1 -• • • -y 2 2n-2 ∩ ± x > 0 ∩ v > ε .
With ε = 1 2 η, we are simply looking at M > r + (a) Ω - >η/2 resp. Ω + >η/2 consists of the space lying above the hyperplane {v = η/2} and below the left (resp. right) tip of the saddle, namely in equations:

Ω ± >η/2 = v > η/2 ± x > 0 v < x 2 -y 2 1 -• • • -y 2 2n-2 .
(b) Ω - >η/2 is the small nose as in (a) but Ω + >η/2 consists of the other side, i.e. of the (rather bigger) space lying inside v > η/2 left to M + >η/2 , namely in equations:

Ω + >η/2 = v > η/2 x > 0 v x 2 -y 2 1 -• • • -y 2 2n-2
.

(c) Symetrically to (b), Ω + >η/2 is the small nose as in (a) but

Ω - >η/2 = v > η/2 x < 0 v x 2 -y 2 1 -• • • -y 2 2n-2
.

(d) Finally, Ω - >η/2 is as in (c) and Ω + >η/2 is as in (b). The last subcase (d) cannot occur, because it is ruled out by property (iii) of Proposition 5.3, which holds on the super-level set (•) > r + λ by the inductive assumption.

Secondly, we assume that c 2 = c 1 . Then there can occur a subcase (a') very similar to (a), in which c 2 = c 1 , so that Ω - >η/2 and Ω + >η/2 belong to the same enclosed relatively compact domain. But with c 2 = c 1 , no subcase similar to (b)or to (c) -can occur, because M - >η/2 ⊂ ∂Ω + >η/2 -or M + η/2 ⊂ ∂Ω - >η/2would then bound the same relatively compact domain from its both sides, but we already know from the beginning of the proof, that one side at least must always contain the points at infinity.

Finally, with c 2 = c 1 = c, there remains the following last subcase (unseen previously).

Ω >η/2 = v > η/2 v > x 2 -y 2 1 -• • • -y 2 2n-2 .
As M = ∂Ω M lies in C n with n 2, whence 2n -2 2, there is at least one dimension of y ∈ R 2n-2 which is missing in the left figure above. To view the topology more adequately, coming back to the abstract quadric Q 1 and assuming n = 2, we plan to draw a short movie consisting of the 3-dimensional slices

v = ε ′ ∩ R 2n Q 1 , where ε ′ = 2 3 η, 1 2
η, 0, -1 2 η. Recall that we are interested in the connected components of the super-level sets together with a geometrically clear connectedness property yield unique holomorphic extension to

v > ε ∩ R 2n Q 1 = ε ′ >ε v = ε ′ ∩ R 2n Q 1 .
Ω c 2 > r + λ with ∂ Ω c 1 > r + λ = M c 1 > r + λ ∪ N c 1 r + λ ∪ R c 1 r + λ and ∂ Ω c 2 > r + λ = M c 2 > r + λ ∪ N c 2 r + λ ∪ R c 2
For i = 1, 2, denote by R c i ,k r + λ , 1 k k λ,c i , the connected components of R c i
Rind R c 1 r + λ , η Rind R c 2 r + λ , η V δ M * > r - λ r - λ Ω c 1 > r + λ Ω c 2 > r + λ .
In sum, we have got unique holomorphic extension to

V δ M * > r - λ r - λ Ω * > r - λ .
To establish (iv) of Proposition 5.3 at (•) > r - λ , it suffices to show (iii), which is checked to be equivalent. We observe that, for logical reasons only, a given region R c The proof of B in case k λ = 1, subcase (a) is complete. Subcase (a') involves only minor differences.

Subtracting process.

We now summarize the discussion of subcase (b), focusing only on topological aspects and dropping the formal considerations about holomorphic extensions. For an adequate three-dimensional illustration, think of a smoothly cut cylindrical piece of modelling clay in which a thin finger drills a hole. , i = 1, 2, which enter C η as the two caps of the third pic of Figure 19.

By descending the interval ( r λ , r + λ ) up to (•) > r λ , we get two regions R c i ,1 r λ , i = 1, 2, that touch at p λ , namely the left cone and the exterior of the right cone in the second pic of Figure 19.

While descending further to (•) > r λ -ε , with ε > 0 very small, the left cone does merge with the right (white) cone. Observe that the points of this (white) cone may be joined continuously to points of the (white) right cap of the first pic, which by hypothesis lies outside Ω c 2 > r + λ , hence in the same connected component as the points at infinity. Consequently, we cannot prolong the left domain Ω c 1 > r + λ so that its prolongation contains the left cone in the slice {v = 0} (third pic), because no admissible prolongation would enjoy the relative compactness (i) of Proposition 5.3. Hence we have no other choice except to suppress Ω c 1 > r λ when attaining (•) > r λ . We then get a new domain Ω * > r λ defined as Ω c 2 > r λ minus the closure of Ω c 1 > r λ (subtraction process), which is checked to be relatively compact in C n . This domain then descends as a uniquely defined domain Ω *

Table of contents 1 .

 1 The Hartogs extension theorem and the method of analytic discs . . . . . . . . . . . . 1. 2. Preparation of the boundary and unique extension . . . . . . . . . . . . . . . . . . . . . . . . . 5. 3. Quantitative Hartogs-Levi extension by pushing analytic discs . . . . . . . . . . . . . . 9. 4. Filling domains outside balls of decreasing radius . . . . . . . . . . . . . . . . . . . . . . . . . 13. 5. Creating domains, merging and suppressing connected components . . . . . . . . 20. 6. The exceptional case k λ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.

Fig. 1 : 2 r 3

 123 Fig. 1: Filling the domain, creating, merging and suppressing components

  which meet; and suppressing one superfluous component Ω c ′ 1

Fig. 2 :

 2 Fig. 2: Preparing the boundary

2. 4 .

 4 Unique holomorphic extension. If U ⊂ C n is open, O(U) denotes the ring of holomorphic functions in U. Definition 2.5.

Fig. 3 :

 3 Fig. 3: Checking connectedness of ΩM ∩ V(∂Ω)

Fig. 4 : 1 ε ⊂ C 2 0 x2Lemma 3 . 2 .

 41032 Fig. 4: Two views of the standard Hartogs figure H 2-1 ε ⊂ C 2

r TpS 2n- 1 rFig. 5 : 1 r( 1 r

 1511 Fig. 5: Relevance of the Levi-Hartogs figure Near a point p ∈ S 2n-1 r (left figure), all copies of C n-1 (in green) which are parallel to the complex tangent plane T c p S 2n-1 r and which lie above the real plane T p S 2n-1 r

r

  , which in addition sends the half-parallelepiped (open) part outside B n r .

Fig. 6 :

 6 Fig. 6: Size of the piece of (green lemon) rind By a (radial) rind of thickness η > 0 around an open region R ⊂ S 2n-1 r

5 .

 5 Its shape, when projected onto T p S 2n-1 r , can either be (approximately) a parallelepiped |z ′ | < c δ, |x n | < c δ , as in the figure, or say, a ball ||z ′ || 2 + |x n | 2 1/2 < c δ ; only the scaling constant c changes.

2n- 1 Corollary 3 . 6 ..

 136 and by controlling monodromy (see rigorous arguments below) we deduce: By means of a finite number C r δ 2n-1 of Levi-Hartogs figures, O S r+δ r extends holomorphically to the slightly deeper spherical shell S r+δ r-c δ 2 r This application could seem superfluous, because large analytic discs with boundaries contained in S r+δ r would yield holomorphic extension to the whole ball B n r+δ in one single step. However, in our situation illustrated by Figure 1, when intersecting S 2n-1 r

Proposition 3 . 7 .

 37 Let R ⊂ S 2n-1 r (with r > 1 and n 2) be a relatively open set having C ∞ boundary N := ∂R and let δ > 0 with 0 < δ < < 1. Then holomorphic functions in the open piece of shell (a one-sided neighborhood of R ∪ N):

  do extend holomorphically to a rind of thickness c δ 2 r around R by means of a finite number C area(R) δ 2n-1 of Levi-Hartogs figures.

Fig. 7 :

 7 Fig. 7: Semi-global extension from a pseudoconcave piece of shell

Fig. 8 :

 8 Fig. 8: Filling the domain from the farthest point

  do extend holomorphically and uniquely to Ω >r by means of a finite number C rκ δ 2n-1 rκ-r η of Levi-Hartogs figures.

Lemma 4 . 8 .

 48 The following intersection of two open sets is connected:

Fig. 11 :

 11 Fig. 11: Joining a point p ′ of the lower rind to the connected upper rind

  is onto, and this holds true since by assumption M ∩ r λ < ||z|| < r λ+1 contains no critical points of r(z)| M . Then (b) follows from this transversality (a). Next, consider the Euclidean metric (v, w) := 2n k=1 v k w k on C n ≃ R 2n , which induces a Riemannian metric (•, •) M on M, a nondegenerate positive bilinear form on T M. The gradient ∇(r| M ) of r(z)| M is the vector field on M defined by requiring that ∇(r| M ), X M = d r| M (X) for all C ∞ (locally defined) vector fields X on M. Let D := 2 Re n k=1 z k ∂ ∂z k be the radial vector

  inside the 2-dimensional plane Π z generated by X D (z) = 0 and by D(z) = 0. If, contrary to what we want, X D (z) would belong to Ker d r| M = T z S 2n-1

Proposition 5 . 3 .

 53 Fix a radius r satisfying r λ < r < r λ+1 for some λ with 1 λ κ -1 and let M c >r , c = 1, . . . , c λ , denote the collection of connected components of M ∩ {||z|| > r}. Then:(i) each M c >r bounds in {||z|| > r} a unique domain Ω c >r which is relatively compact in C n ; (ii) the boundary in C n of each Ω c>r , namely:∂ Ω c >r = M c >r ∪ N c r ∪ R cr consists of M c >r together with some appropriate union N c r of finitely many connected components of N r = M ∩ {||z|| = r} and with an appropriate region R c r ⊂ S 2n-1 r delimited by N c r ; (iii) two such domains Ω c 1 >r and Ω c 2 >r , associated to two different connected components M c 1 >r and M c 2 >r of M >r , are either disjoint or one is contained in the other; (iv) for c 1 = c 2 , the regions R c 1 r and R c 2 r are either disjoint or one is contained in the other, while their boundaries N c 1 r and N c 2 r are always disjoint; (v) for each c = 1, . . . , c λ , every function f holomorphic in V δ M >r >r has a restriction to V δ M c >r >r which extends holomorphically and uniquely to Ω c >r by means of a finite number of Levi-Hartogs figures.

Fig. 12 :Theorem 5 . 4 .

 1254 Fig. 12: Possible topologies of the cut out hypersurfaces M >rAs a direct application, we may achieve the proof of our principal result. Theorem 5.4. Under the precise assumptions of Theorem 2.7, holomorphic functions in V δ (M) do extend holomorphically and uniquely to Ω M by means of a finite number of Levi-Hartogs figures:∀ f ∈ O V δ (M) ∃ F ∈ O Ω M ∪ V δ (M) s.t. F V δ (M ) = f.

λ+1, c = 1 , 2 j

 12 . . . , c λ , as well as the corresponding holomorphic extensions, have been constructed. Then prolong the domains (without topological change) as Ω c > r + λ , c = 1, . . . , c λ , up to r + λ = r λ + η 2 and fill in the conquered territory by means of a finite number of Levi-Hartogs figures. B: Jumping across singular radii and changing the domains. Restarting at r + λ with the domains Ω c > r + λ , c = 1, . . . , c λ , distinguish three cases as follows.Remind from §2.3 that M is represented by v = 1 j k λ x 2 j -1 j 2n-k λ -1 yin suitable coordinates (x, y, v) centered at p λ , where k λ is the Morse coindex of r(z)| M at p λ .(I) Firstly, assume k λ = 0, namely z → r(z)| M has a local maximum at p λ , or inversely, assume k λ = 2n -1, namely z → r(z)| M has a local minimum at p λ . This is the easiest case, the only one in which new domains can be born or die, locally.

Fig. 13 :

 13 Fig. 13: GettingV δ M c

  shrink conveniently the thickness of the last extensional rind, as in the proof of Proposition 4.5). Thus, by piling up r - λ+1 -r + λ η rinds and by using a finite number C rκ δ 2n-1 r - λ+1 -r + λ η of Levi-Hartogs figures, we get unique holomorphic extension to (5.10)

13 . 5

 135 Topology of horizontal super-level sets in the complement of quadrics.Simultaneously to the proof, we provide an auxiliary elementary study.Let n ∈ N with n 2, let k ∈ N with 0 k 2n -1, let x = (x 1 , . . . , x k ) ∈ R k , let y = (y 1 , . . . , y 2n-1-k ) ∈ R 2n-1-k , let v ∈ R,and in R 2n equipped with the coordinates (x, y, v), consider the quadric of equation (denote by Q k . The coordinate v playing the rôle of r(z)r( p λ ) near a singular radius r λ having Morse coindex k λ , we want to understand how the topology of the super-level sets v > ε ∩ R 2n \Q k(which relate to the possible domains Ω c >r for r close to r λ ) do change when the parameter ε descends from a small positive value to a small negative value.

Fig. 15 :

 15 Fig. 15: Growing of superlevel domains near a local maximum or minimum

Fig. 16 :

 16 Fig. 16: Two distinct Hartogs-Levi fillings at a point of Morse coindex 2n -1

  . , R c,k λ,c r + λ , we may achieve the Hartogs-Levi filling exactly as in A, because r(z)| M is regular in V δ N c,j r + λ , for every j such that N c,j r + λ is contained in the boundary of each of these other components. Hence it remains only to discuss what is happening in a neighborhood of the single component R c,1 r + λ , and especially near p λ . For the disposition of Ω c > r + λ ∩ C η , or equivalently of R c,1 r + λ ∩ C η , two cases occur. Let v, x 1 , . . . , x 2n-1 be the Morse coordinates centered at p λ . (a) As illustrated by the left figure above, Ω c > r + λ ∩ C η consists of the space 7 lying above v = η/2 and above v = x 2 1 + • • • + x 2 2n-1 , a cap-shaped space which is clearly connected; the region R c,1 r + λ is then diffeomorphic to a small (2n -1)-dimensional ball. (b) As illustrated by the right figure above, Ω c > r + λ ∩ C η consists of the space

  The union of this open set together with (5.10) contains a unique well defined domain Ω c > r - λ with the property that the passage from R c, as illustrated by the right diagram above, whence N c > r - λ has one less connected component, because the (2n -2)-sphere N c,1 > r λ +ε drops when ε < 0. The properties that two different domains Ω c 1 or one is contained in the other is easily seen to be inherited by Ω c 1 > r - λ and Ω c 2 > r - λ : it suffices to distinguish two cases: c 2 = c and c 1 = c, or c 2 = c and c 1 = c; to look at (a) or (b) and then to conclude.

Fig. 17 :

 17 Fig. 17: Sliced view of the growing of the two possible domains in case 2 k λ 2n -2 So let M c > r + λ be the single connected component of M ∩ ||z|| > r + λ that enters C η . The corresponding domain Ω c > r + λ

1 6. 1 .Fig. 18 :

 1118 Fig. 18: Slices and superlevel sets at a Morse point of coindex k λ = 1

λ∩

  C η . By descending induction through A and B, we are given two domains of holomorphic extension Ω c 1 > r + λ and Ω c 2 > r + λ whose boundary contains M - >η/2 and M + >η/2 , respectively. Firstly, we assume that c 2 = c 1 . Since each one of the two pieces of hypersurfaces M - >η/2 and M + >η/2 has two sides, there are 2 × 2 = 4 subcases to be considered for the relative disposition of Ω - >η/2 := Ω c 1 > r + λ ∩ C η and of Ω + >η/2 := Ω c 2 > r + λ ∩ C η , with c 2 = c 1 .

  (e) Ω >η/2 := Ω c > r + λ ∩ C η consists of the space lying above v = η/2 and above the saddle, namely

Fig. 21 : 1 6. 2 .

 2112 Fig. 21: Sliced view of the growing of the external domain in subcase (e) of k λ = 1

  ii) of Proposition 5.3, all the other properties also holding true on (•) > r + λ . We remind that the other domains Ω c > r + λ for c = c 1 and c = c 2 with 1 c c λ do pass regularly through r λ up to (•) > r - λ , thanks to A.

λ

  and by N c i ,j r + λ , 1 j j λ,c i , with j λ,c i k λ,c i , the components of N c i r + λ . We do the numbering so that R c 1 ,1 r + λ (resp. R c 2 ,1 r + λ ) enters C η to the left (resp. right), together with N c 1 ,1 r + λ (resp. N c 2 ,1 r + λ ), as illustrated by Figure 17. As in the case k λ = 2n -1, for i = 1, 2, by means of extensional rinds that are symmetric around the other components R c i ,2 r + λ , . . . , R c i ,k λ,c i r + λ , we may achieve the Hartogs-Levi filling exactly as in A, because r(z)| M is regular in V δ N c i ,j r + λ , for every j such that N c i ,j r + λ is contained in the boundary of each of these other components. Hence it remains only to discuss what is happening in a neighborhood of the two components R c i ,1 r + λ , i = 1, 2, and especially near the saddle point p λ . While descending from r + λ to r - λ , the two regions R c 1 a single connected region contained in S 2n-1 r - λ that we will denote by R * r - λ , see the right Figure 17. In Morse theory ([31, 19]), one speaks of attaching a one-cell, since in the merging process, the two regions are essentially joined by means of a (thickened) segment directed along the x-axis. It follows that the two hypersurfaces M c 1 > r + λ and M c 2 > r + λ do merge as a connected hypersurface M * > r - λ containing them, and furthermore, that the two domains Ω c 1 up to the slightly deeper super-level set (•) > r - λ , merging as a uniquely defined domain Ω * > r - λ which is relatively compact in C n and which contains R * r + λ in its boundary ∂ Ω * > r + λ . As c 2 = c 1 , the new number of domains in the interval ( r λ-1 , r λ ) is lowered by a unit, i.e. c λ-1 = c λ -1 (if c 2 = c 1 as in (a'), the number would not change, i.e. c λ-1 = c λ ). For i = 1, 2, let f c i r + λ denote the restriction of f ∈ O V δ M >r >r to V δ M c i > r + λ > r + λ . By descending induction through A and B, f c i r + λ extends holomorphically and uniquely to Ω c i > r + λ . Then both functions do extend holomorphically and uniquely to V δ M * > r - λ > r - λ , since they coincide with f near p λ . We then introduce the two extensional rinds Rind R c i r + λ , η , drawn in the right Figure 17. Two applications of Proposition 3.7

;•∩∩.

  r + λ for c = c 1 and c = c 2 can:• be disjoint from R c 1 r + λ and also disjoint from R c 2 r + λ be contained in R c 1 r + λ or (exclusive "or") in R c 2 C η in the level set v = η 2 ∩ C η , which is impossible since N C η , not more.It follows in all cases that N c r+ λ = ∂ R c r + λ is disjoint from C η ,hence it lies in r - λ ||z|| r + λ C η . Consequently, the regular flow of ∇ r M ||∇r M || on M ∩ r - This information suffices now to check that (iii) and (iv) of Proposition 5.3 are transmitted to (•) > r - λ , just for logical reasons.

Fig. 22 :

 22 Fig. 22: Three-dimensional view of subcase (b) at a point of Morse coindex k λ = 1 As in §5.6, in C η , there enter exactly two domains Ω c i > r + λ

  a corresponding connected region R * r - λ approximately equal to R c 2 ,1 r λ minus the closure of R c 1 ,1 r λ whose boundary contains a connectedd N * r - λ (bottom right Figure 21), obtained by merging N c 1 ,1 r λ with N c 2 ,1 r λ . The last subcase (e) above is topologically similar to what happens in §5.15, hence the proof of Proposition 5.3 is complete.

3.1. The classical Hartogs figure. Local

  Hartogs phenomena can now enter the scene. They involve translating ("pushing") analytic discs and they will provide small, elementary extensional steps to fill in Ω M .

	§3. QUANTITATIVE HARTOGS-LEVI EXTENSION
	BY PUSHING ANALYTIC DISCS
	-Hartogs figures ( §3.3),
	without using neither the Martinelli kernel, nor solutions of an auxiliary ∂ prob-
	lem.

Because in the sequel, the union U 1 ∪ U 2 would sometimes be a rather long, complicated expression (see e.g. (3.9)), hence uneasy to read, we will also say that O(U 1 ) extends holomorphically and uniquely to U 2 .

We let the letter c (resp. C) denote a positive constant < 1 (resp. > 1), absolute or depending only on n, which is allowed to vary with the context.

As suggested by this sliced union, to conceptualize these 4-dimensional (in case n = 2) super-level sets, it suffices to pile up intuitively the images of the corresponding movie. the third picture, namely the slice ε ′ = 0, shows a not allowed situation: the left cone does bound two regions from its two sides, contrary to the a priori unique relatively compact domain Ω c 1 r λ ⊂ ||z|| > r λ we are seeking to construct, when starting from Ω c 1 r + λ . The trick is then to suppress the (blue) small slice, or equivalently to subtract it from the (black) large slice which contains it. Then the black winning slice continues to grow up to v = -η/2 (fourth picture). The complete discussion follows in a while.

Finally, here is the (simpler) movie of (e).


An Optical Flow Method

Applied to Co-Registration of Remote Sensing Images: Example for SAR/SAR, SAR/LIDAR, SAR/Optical Images of BIOSAR 2010 Campaign

AN OPTICAL FLOW METHOD APPLIED TO CO-REGISTRATION OF REMOTE SENSING IMAGES: EXAMPLE FOR SAR/SAR, SAR/LIDAR, SAR/OPTICAL IMAGES OF BIOSAR 2010 CAMPAIGN

Elise Colin-Koeniguer (1) (1) Onera, Chemin de la Hunière et des Joncherettes, 91123 PALAISEAU Cedex FRANCE Email: elise.koeniguer@onera.fr

ABSTRACT

This article proposes an optical flow type method for coregistration of forest remote sensing images. The principle of the algorithm called GeFolki is first explained. Results are shown on the images of the BioSAR 3 campaign, for the production of SAR interferograms, the coregistration a SAR and LIDAR image, and the coregistration an optical image and SAR image.

The advantages of such an algorithm over conventional algorithms are explained. Finally, we propose various applications within the operating data for future BIOMASS mission: massive interferometry, ground truth production, upscaling by fusion of LIDAR and SAR data, and image mining.

MOTIVATION

In remote sensing, data are now ever growing. To benefit from this wealth of information, and to make the fusion of several sensors, a first step involves the common mapping of all the images available on the same site, as shown in Figure 1. This is the stage called coregistration. For all our forest images, we want to have a tool able to superimpose the different images of the same site with a tool that is fast enough to handle large temporal stacks or large volumes. Also, in this article we propose the use of an algorithm based on the classical optical flow framework, called GeFolki.. In the next section, we give the brief description of how the algorithm works. Then, we show the results obtained on the BioSAR 3 images. The performance in terms of precision and speed of execution are discussed quantitatively. Finally, we propose several different application frameworks in the case of evaluation of forest biomass.

Figure 1. Example of composition using all products

that can be useful for biomass estimation: from left to right: Optics, LIDAR, L-band POLSAR, P-band POLSAR and P-band InSAR.

GENERAL DESCRIPTION OF THE ALGORITHM

The algorithms for optical flow provide a displacement map between two images. Generally, they rely on the calculation of an optical flow equation, which is derived from a data model wherein:

-The images of one frame to the next are identical in terms of intensity variations. This hypothesis is called the brightness constancy hypothesis.

-Displacements to look for are small enough to allow the use of Taylor series expansion. This can be called the small motion hypothesis. Although these assumptions are absolutely not respected in the context of remote sensing images, we aim still to be inspired by these methods.

The method we have developed is called GeFolki. This acronym is a historical result of the following acronyms: Folki is an initial algorithm of Optical Flow, inspired by the Lucas-Kanade algorithm [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF], developed by the G. Le Besnerais and F. Champagnat at Onera [START_REF] Besnerais | Dense optical flow by iterative local window registration[END_REF]. EFolki is an extended version, developed and implemented on GPU by Aurélien Plyer during his PhD thesis [START_REF] Plyer | Massively parallel Lucas Kanade optical flow for real-time video processing applications[END_REF]. GeFolki is devoted to the new version of remote sensing images for geoscience.

As for the local optical flow algorithm originally developed by Lucas and Kanade, Gefolki is based on the minimization of the following criterion of Eq 1:

( ) ∑ ( ) (1)
where is a local neighborhood of the pixel at position x, u(x) is the desired displacement, I 1 and I 2 are the master and slave images to coregister. Here, the key feature of GeFolki lies in the definition of the functions f 1 and f 2 . These two functions convert the intensities so that the brightness constancy assumption is satisfied for transforms of the two images.

The functions that we propose are defined as follows:

R is a ranking function. This rank function consists in replacing the intensity value I(x) by the number of pixels in the neighborhood of x with an intensity lower than I(x). f is a filter function preserving edges and called rolling guidance filter, described in [START_REF] Zhang | Rolling guidance filter[END_REF] and g is a function of local contrast inversion decision.

Figure 2. Example of application of the inversion contrast decision on optical and radar images

This function g corresponds to a rapid test to implement and decide whether the intensity of the slave image has to be inverted or not. This function allows obtaining intensity variations with similar contrasts in both images, whereas this is not necessarily the case on the initial images, as shown in Fig 2.

Finally, a pyramid-solving approach enables to respect the assumption of small motion, as conventionally done in the optical flow algorithms dedicated to relatively large displacements.

RESULTS

Several results are obtained images of the site of the BIOSAR3 campaign. The optical images are from Lantmäteriet, and LIDAR images were provided by SLU and the FOI. They are represented in Fig. 3. The results obtained enable to qualitatively show a significant improvement compared to coregistration final products obtained by a fine geocoding method.

To strengthen the quality of products quantitatively, we proposed an accuracy assessment method based on the detection of the trees in the master image and in the slave image, and then matching them, as in the example of the Fig. 5 containing 5 trees.

In this case, it is possible to make statistics on the distances obtained between the detected pairs of trees. These statistics show that we have a precision of about 10 pixels in the case of geocoding against an accuracy of about less than two pixels in the case of GeFolki.

Regarding the speed of execution, we also compared GeFolki to a conventional method based on mutual information for the coregistration heterogeneous images [START_REF] Kroon | MRI modalitiy transformation in demon registration[END_REF]. The execution speeds are faster by a factor of 10.

The transition to an implementation on GPU saves an additional factor 100.

APPLICATIONS FOR BIOMASS

Interferogramm production

One of the first possible applications is the production of SAR interferograms. Compared to other existing methods, it has the advantage of being fast and not requiring any auxiliary data. Moreover, it is a dense method, that is to say, the deformation can be calculated at all points of the image, unlike the methods that evaluate this deformation on specific tie-points, or on a pixel subgrid. This advantage is particularly important in the case of airborne data and high resolution images, which undergo large deformations and that difficult to correct, because of the instability of the trajectory path.

Ground Truth production

In the context of the use of machine learning methods, it is necessary to label a number of data in order to constitute the training data. If the classification that is desired is made pixel by pixel, then the associated labeling must also be done for each pixel of the image. One way of doing so is to start from a ground truth given by another image, and coregister it with the image that is to be classified. This is what we proposed to do to classify a polarimetric SAR image into different vegetation classes. The ground truth is generated from the volumetric density raster files of each species of trees, provided by the SLU. These layers themselves were obtained from SPOT image and in situ field measurement. The resulting layers are created at much lower resolutions to ours (30m against 80 cm for the SAR image). However, it is possible to coregister the images all the same, and get a set of labels like in Fig. 7.

Fusion at the pixel level

Another interesting application relates to the fusion of images from different sensors. For the assessment of biomass, it is the PhD subject of G. Brigot [START_REF] Brigot | Fusion of LIDAR and POLINSAR images for forest vertical structure retrieval From EUSAR[END_REF]. The main idea is to learn the existing inference function between POLINSAR parameters and LIDAR profile, by machine learning. This requires selecting a radar feature that is most correlated possible to the LIDAR feature, for each pixel of the image, after the coregistration process.

We chose as POLINSAR feature, a number of geometrical parameters to describe the coherence shape.

The LIDAR feature or associated label is the percentage points of the return back to the sensor, according to the elevation.

Figure 8 Upscaling process: the inference function between a POLINSAR feature, here the coherence shape, and the LIDAR vertical profiles is learnt through a machine learning method.

To implement the machine learning process, the first step is to match a number of features. This is what was done by the coregistration of ALS data, at 10mx10m resolution, provided by SLU, with our POLINSAR radar data.

Image mining

Recently, we also have developed an image mining algorithm based on the same criterion of similarity that GeFolki.

From this algorithm, we are for example able to find a high resolution image of TerraSAR-X, inside the Sentinel image of the same area, as illustrated in Fig. 10.

In a first step, we are simply trying to center the smallest image on the most probable center pixel from the Sentinel image. Secondly, the residual deformations can be corrected from the classic GeFolki algorithm.

This image mining stage could be an automatic initialization of the flow calculation between any two images.

Figure 10 Two steps of an image mining process: in the first step the best position of the TSX image is found inside the Sentinel 1 image. In the second step, results are improved thanks to GeFolki

CONCLUSION

We have shown how we managed to adapt a method of optical flow coregistration in case of heterogeneous remote sensing data registration. The spatial accuracy is improved by a factor of 5, and execution speed by a factor of 10. Especially, the algorithm offers many advantages such as dense evaluation of deformation between two images, and the independence to the auxiliary data. This allows to produce interferograms under difficult conditions, but also to consider new applications such as ground truth production for machine learning, fusion of heterogeneous images at the pixel-wise level, and image mining.

AKNOWLEDGMENTS AND CREDITS

All data used in this paper come from various institutions:

-LIDAR data have been provided by Johan Fransson from SLU, Swedish University of Agricultural Sciences, for whom we are deeply grateful. All LIDAR images have been processed by SLU. We also thank Lars Ulander from FOI for his help on the DEM.

-Polarimetric SAR data set over Remningstorp is provided from the BIOSAR 2010 campaign funding by ESA, acquired by Onera.

-The optical image over Sweden has been provided by Lantmäteriet.

Figure 3 .

 3 Figure 3. Description of the data set from the BIOSAR 3 campaign: radar, LIDAR and optics image.

Figure 4 .

 4 Figure 4. A small example of the improvement brought by the coregistration algorithm in comparison to the geocoding coregistration.

Figure 5 .

 5 Figure 5. P-band Interferogramm in HSV colored compositionThus, we generated the P-band interferogram of the area, as shown in Fig .5. The orbital fringes were calculated from a polynomial regression of order 2 of the fringe pattern thus obtained. Similarly, we generated the L-band interferogram, which proved even more difficult because of very fast variations of the interferometric phases, caused by a low ambiguity height. The result is shown in Fig.6

Figure 6 .

 6 Figure 6. L-band Interferogramm in HSV colored composition

Figure 7 .

 7 Figure 7. Example of a ground truth -For each pixel of the SAR image corresponds a label.

Figure 9

 9 Figure 9 Some results of the upscaling process after the coregistration of all LIDAR raster data and SAR images.

Figure 9

 9 Figure 9 illustrates a number of learning outcomes obtained by SVM: four pixels of different radar images, profiles learned from the radar are shown in blue, while the LIDAR profiles considered as ground truth are represented in red color.

Guillaume Brigot, is a PhD student jointly supervised by Elise Koeniguer in Onera and Marc Simard in JPL. His PhD is supported jointly by TOTAL and Onera.