Related papers: Isometric disks are holomorphic

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $\Omega_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex…

This paper shows that, in dimensions two or more, there are no holomorphic isometries between Teichm\"uller spaces and bounded symmetric domains in their intrinsic Kobayashi metrics.

We study isometric maps between Teichm\"uller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but…

In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon \Omega\to\Omega'$…

In this paper, we discuss a rigidity property for holomorphic disks in Teichm\"uller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds.…

In this article, we study the Kobayashi isometries of 2-dimensional complex manifolds having a finite Carath\'eodory universal set. In particular, we prove that the Kobayashi isometries of these complex manifolds are (anti)holomorphic.

We first give an exposition on holomorphic isometries from the Poincar\'e disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e disk with a complex unit ball. As an application, we provide an example of proper…

A new metric on the open 2-dimensional unit disk is defined making it a geodesically complete metric space whose geodesic lines are precisely the Euclidean straight lines. Moreover, it is shown that the unit disk with this new metric is not…

Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant…

We prove that the every quasi-isometry of Teichm\"uller space equipped with the Teichm\"uller metric is a bounded distance from an isometry of Teichm\"uller space. That is, Teichm\"uller space is quasi-isometrically rigid.

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We…

Pulling back complex structures along a branched covering induces a holomorphic isometric embedding of Teichm\"uller spaces. We show that for dimension at least $2$, all isometric embeddings arise from branched coverings. This generalizes a…

We show the (anti)holomorphicity of smooth Kobayashi isometries of the diamond, the domain defined as $\triangle:=\{z\in\mathbb C^2:|z_1|+|z_2|<1\}$. Additionally, we discuss the problem of uniqueness of real geodesics, left inverses and…

We study the class of holomorphic and isometric submersions between finite-type Teichm\"uller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map $\mathcal{T}_{g,n} \rightarrow…

Let $S$ be a hyperbolic Riemann surface. In a finite-dimensional Teichm\"uller space $T(S)$, it is still an open problem whether the geodesic disk passing through two points is unique. In an infinite-dimensional Teichm\"uller space it is…

We study holomorphic isometries between bounded symmetric domains with respect to the Bergman metrics up to a normalizing constant. In particular, we first consider a holomorphic isometry from the complex unit ball into an irreducible…

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichm\"{u}ller space, degenerating to the Riemann surface where it is pinched. We show there is a…

We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold with…

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon \Omega\to\Omega'$ between bounded symmetric domains $\Omega$ and $\Omega'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries…

This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…