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Register a new userVisibility of Kobayashi geodesics in convex domains and related properties
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Let $Dsubset mathbb C^n$ be a bounded domain. A pair of distinct boundary points ${p,q}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.
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In this paper we study the following slice rigidity property: given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $mathcal F$ of $M$, when is it true that every holomorphic map $F:Mto N$ which maps isometrically every complex geodesic of $mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $mathcal F$ contains a given point of $overline{M}$ and $mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.
In this paper we study the global geometry of the Kobayashi metric on convex sets. We provide new examples of non-Gromov hyperbolic domains in $mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon ewline -vex, bounded and unbounded. Our first aim is to prove that if $Omega$ is a bounded weakly linearly convex domain in $mathbb{C}^n,,ngeq 2,$ and $S$ is an affine complex hyperplane intersecting $Omega,$ then the domain $Omegasetminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $Omegasetminus S,$ where $S$ is relatively closed in $Omega$ and $Omega$ is a convex domain, depends only on that how $S$ looks near the boundary, i.e., whether $S$ near $partialOmega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
We consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of A_g. Both Shimura curves and Teichmueller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field.
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.
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.
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