相关论文: On isometries of the Kobayashi and Carath\'{e}odor…

The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carath\'eodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains,…

We show that if $Y_j\subset \mathbb{C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary for $j=1,\dots,q$, and $X_j\subset \mathbb{C}^{m_j}$ is a bounded convex domain for $j=1,\ldots,p$, then the product domain $\prod_{j=1}^p…

Let $\1$ and $\2$ be $\s$ domains in $\Cn$ and $f: \1 \rt \2$ an isometry for the Kobayashi or Carath\'eodory metrics. Suppose that $f$ extends as a $C^1$ map to $ \bar \om_1$. We then prove that $f|_{\partial \1}: \partial \1 \rt \partial…

In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carath\'{e}odory metric in some planar domains of finite connectivity greater than $2$, and hence resolve a problem posed by…

For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with…

We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and…

We study the rigidity of maps between bounded symmetric domains that preserve the Carath\'eodory/Kobayashi distance. We show that such maps are only possible when the rank of the co-domain is at least as great as that of the domain. When…

We prove that a strongly pseudoconvex domain with noncompact group of Kobayashi/Royden metric isometries must be biholomorphic to the unit ball.

We study the Carath\'{e}odory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly incisive results are obtained for convex domains.

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carath\'eodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is…

In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. "Visibility" in this context is reminiscent of visibility, seen in negatively curved…

Given a pair of smoothly bounded domains $D_1, D_2 \subset \mathbb C$, the purpose of this note is to obtain an inequality that relates the Carath\'{e}odory metrics on $D_1, D_2, D_1 \cap D_2$ and $D_1 \cup D_2$.

Consider a holomorphic map $F: D \to G$ between two domains in ${\mathbb C}^N$. Let $\mathcal F$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal F$. This paper is…

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $\Omega\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with…

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…

Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex Levi corank one domain with defining function $r$, i.e., the Levi form $\partial \bar {\partial} r$ of the boundary $\partial D$ has at least $(n - 2)$ positive eigenvalues…

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form.…

In this paper we introduce a new class of domains -- log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local…

We prove a Carath\'eodory-type extension of BQS homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carath\'eodory-type extension of geometric…

The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the…