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Isometries in the diamond

Anand Chavan, Włodzimierz Zwonek

Abstract

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 strict convexity of indicatrices.

Isometries in the diamond

Abstract

We show the (anti)holomorphicity of smooth Kobayashi isometries of the diamond, the domain defined as . Additionally, we discuss the problem of uniqueness of real geodesics, left inverses and strict convexity of indicatrices.
Paper Structure (10 sections, 8 theorems, 20 equations)

This paper contains 10 sections, 8 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Motivation and background
  4. Lempert domains. Complex and real geodesics. Left inverses. Indicatrices.
  5. Real geodesics
  6. Property of uniqueness of right inverses and real geodesics
  7. (Infinitesimal) complex isometries
  8. The case of the domain $\triangle$
  9. Left inverses and the formula for $\kappa_{\triangle}$
  10. Smooth Kobayashi isometries of $\triangle$

Key Result

Theorem 1

Let $F:\triangle\to\triangle$ be a $C^1$-smooth Kobayashi isometry. Then $F$ is holomorphic or antiholomorphic, in other words, up to a permutation of variables it is of the form $F(z)=(\omega_1z_1,\omega_2z_2)$, $z\in\triangle$ or $F(z)=(\omega_1\overline{z_1},\omega_2\overline{z_2})$, $z\in\triang

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Corollary 5
  • Proposition 6
  • proof
  • ...and 7 more