Searcharxiv⌕ Search

arXiv subjects

Hervé Gaussier

Publications and source records attributed to Hervé Gaussier.

At least 19 recordsLinked to original sources
arXiv2026

Curvature of hyperbolic complex manifolds

The article addresses the construction and geography of negatively curved metrics on hyperbolic complex manifolds. We introduce a mechanism for constructing complete Kähler metrics with negative bisectional curvature. This applies to some product complex manifolds, thereby resolving a longstanding problem attributed to N. Mok. We then construct projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes $c_1^2/c_2$ realize any $s \in \mathbf{Q} \cap \left( \frac{2}{7}, \frac{2}{3} \right)$. For slopes $s\in \mathbf{Q}\cap \left( \frac{2}{7},\frac{1}{3} \right)$, the corresponding surfaces admit a Hermitian metric with $\text{HSC}<0$, but their Kähler--Einstein metric cannot have $\text{HSC}<0$. We finally construct, for every $s \in \left( \frac{1}{2}, 3 \right)$, a sequence of projective Kobayashi hyperbolic surfaces that do not admit a Hermitian metric of nonpositive holomorphic sectional curvature, whose Chern slopes $c_1^2/c_2$ converge to $s$.

math.DG↗
arXiv2025

Kobayashi hyperbolicity in Riemannian manifolds

We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs. We also prove a Picard theorem for conformal harmonic discs and give some examples of Kobayashi hyperbolic Riemannian manifolds.

math.CV↗
arXiv2023

Smooth equivalence of families of strongly pseudoconvex domains

We establish a smoothness result for families of biholomorphisms between smooth families of strongly pseudoconvex domains, each with trivial biholomorphism group. This is accomplished by considering the Riemannian geometry of their Bergman metrics and proving a result about the smoothness of families of isometries between smooth families of Riemannian manifolds.

math.CV↗
arXiv2023

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

We introduce the notion of locally visible and locally Gromov hyperbolic domains in $\mathbb C^d$. We prove that a bounded domain in $\mathbb C^d$ is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.

math.CV↗
arXiv2022

Abstract boundaries and continuous extension of biholomorphisms

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in ${\mathbb C}^d$, $d \geq 1$. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps.

math.CV↗
arXiv2022

The geometry of domains with negatively pinched Kähler metrics

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive domain. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D'Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. In particular, we show that a convex domain with $C^{2,α}$ boundary is strongly pseudoconvex if and only if it admits a complete Kähler metric with sufficiently tight pinched negative holomorphic sectional curvature outside a compact set.

math.CV↗
arXiv2022

A metric analogue of Hartogs' theorem

In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.

math.CV↗
arXiv2020

Unbounded Kobayashi hyperbolic domains in $\mathbb C^n$

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $\mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $\mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.

math.CV↗
arXiv2019

Homeomorphic extension of quasi-isometries for convex domains in $\mathbb C^d$ and iteration theory

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.

math.CV↗
arXiv2019

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc

Let $Δ\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to Δ$ be a Riemann map and let $\{z_k\}\subset Δ$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$ if and only if there exists a simply connected domain $U\subsetneq \mathbb C$ such that $Δ\subset U$ and $Δ$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if $(ϕ_t)$ is a non-elliptic semigroup of holomorphic self-maps of $\mathbb D$ with Königs function $h$ and $h(\mathbb D)$ contains a vertical Euclidean sector, then $ϕ_t(z)$ converges to the Denjoy-Wolff point non-tangentially for every $z\in \mathbb D$ as $t\to +\infty$. Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but oscillating, in the sense that the slope of the trajectories is not a single point.

math.CV↗
arXiv2018

Asymptotic behavior of orbits of holomorphic semigroups

Let $(ϕ_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Ω$ be the starlike at infinity domain image of the Koenigs function of $(ϕ_t)$. In this paper we completely characterize the type of convergence of the orbits of $(ϕ_t)$ to the Denjoy-Wolff point in terms of the shape of $Ω$. In particular we prove that the convergence is non-tangential if and only if the domain $Ω$ is `quasi-symmetric with respect to vertical axes'. We also prove that such conditions are equivalent to the curve $[0,\infty)\ni t\mapsto ϕ_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $Ω$.

math.CV↗
arXiv2018

A characterization of orthogonal convergence in simply connected domains

Let $\mathbb D$ be the unit disc in $\mathbb C$ and let $f:\mathbb D \to \mathbb C$ be a Riemann map, $Δ=f(\mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence $\{z_n\}\subset Δ$ has the property that $\{f^{-1}(z_n)\}$ converges orthogonally to a point of $\partial \mathbb D$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $\mathbb D$.

math.CV↗
arXiv2018

Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc

We study the backward invariant set of one-parameter semigroups of holomorphic self-maps of the unit disc. Such a set is foliated in maximal invariant curves and its open connected components are petals, which are, in fact, images of Poggi-Corradini's type pre-models. Hyperbolic petals are in one-to-one correspondence with repelling fixed points, while only parabolic semigroups can have parabolic petals. Petals have locally connected boundaries and, except a very particular case, they are indeed Jordan domains. The boundary of a petal contains the Denjoy-Wolff point and, except such a fixed point, the closure of a petal contains either no other boundary fixed point or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of Königs functions using divergence rate and universality of models. Moreover, we construct a semigroup having a repelling fixed point in such a way that the intertwining map of the pre-model is not regular.

math.CV↗
arXiv2018

Horosphere topology

We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains. Among other things, we prove that every univalent map of the unit ball whose image is bounded and convex, extends as a homeomorphism up to the closure. Such a result, relying in an essential way on our theory and on the Gromov hyperbolicity theory, is completely new, dealing with non smooth domains.

math.CV↗
arXiv2017

Smooth equivalence of deformations of domains in complex euclidean spaces

We prove that two smooth families of 2-connected domains in $\cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m \geq 3$, two smooth families of smoothly bounded $m$-connected domains in $\cc$, and for $n\geq2$, two families of strictly pseudoconvex domains in $\cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.

math.CV↗
arXiv2017

A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in $\mathbb C^n$

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be the set of points $ξ$ such that $\lim_{z\to ξ}\|F(z)\|=\infty$. Then we prove that $\mathcal S$ is either empty, or contains one or two points and $F$ extends as a homeomorphism $\tilde{F}:\overline{\mathbb B^n}\setminus \mathcal S\to \overline{D}$. Moreover, $\mathcal S=\emptyset$ if $D$ is bounded, $\mathcal S$ has one point if $D$ has one connected component at $\infty$ and $\mathcal S$ has two points if $D$ has two connected components at $\infty$ and, up to composition with an affine map, $F$ is an extension of the strip map in the plane to higher dimension.

math.CV↗