Invariant holomorphic discs in some non-convex domains
We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.
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arXiv subjects
Hervé Gaussier
Publications and source records attributed to Hervé Gaussier.
Holomorphic motions and complex geometry
We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n .
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Algebraic embeddings of smooth almost complex structures
The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F.Bogomolov: can one embed every compact holomorphic manifold as a C infinity smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?
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Positivity and completeness of invariant metrics
We present a method for constructing global holomorphic peak functions from local holomorphic support functions for broad classes of unbounded domains. As an application, we establish a method for showing the positivity and completeness of invariant metrics including the Bergman metric mainly for the unbounded domains.
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Gromov hyperbolicity of strongly pseudoconvex almost complex manifolds
Let $D=\{ρ< 0\}$ be a smooth relatively compact domain in an almost complex manifold $(M,J)$, where $ρ$ is a smooth defining function of $D$, strictly $J$-plurisubharmonic in a neighborhood of the closure $\overline{D}$ of $D$. We prove that $D$ has a connected boundary and is Gromov hyperbolic.
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Quasi-isometries in strongly convex domains
We provide examples of quasi-isometries for strongly convex domains in $\mathbb C^n$ endowed with their Kobayashi distance.
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On the Gromov hyperbolicity of convex domains in Cn
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.
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Levi-flat filling of real two-spheres in symplectic manifolds (I)
Let (M,J,w) be a manifold with an almost complex structure J tamed by a symplectic form w. We suppose that M has complex dimension two, is Levi convex and has bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M may be foliated by the boundaries of pseudoholomorphic discs.
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Symmetries of partial differential equations
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.
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Nonalgebraizable real analytic tubes in C^n
We give necessary conditions for certain real analytic tube generic submanifolds in C^n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C^n. During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in C^n passing through the origin, having a defining equation of the form v = ϕ(y), where (z,w)= (x+iy,u+iv) \in C^{n-1} \times C. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields \partial_{z_1}, ..., \partial_{z_{n-1}}, \partial_w. Then M is locally algebraizable only if every second derivative \partial^2_{y_ky_l}ϕis an algebraic function of the collection of first derivatives \partial_{y_1} ϕ,..., \partial_{y_m} ϕ.
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