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On left inverses in the Lempert theorem

Włodzimierz Zwonek

TL;DR

The paper investigates left inverses to complex geodesics in Lempert domains, introducing a derivative-based construction of Lempert left inverses from given left inverses and establishing a one-parameter family connecting different inverses. It analyzes regularity and boundary-extension obstructions, providing explicit examples in the bidisc and symmetrized bidisc, including royal and flat geodesics, and develops a Burns-Krantz rigidity framework for ellipsoids. The work also discusses the role of dual functions in building left inverses, and addresses invariance under holomorphic automorphisms, highlighting both positive results and notable limitations. Together, these results deepen understanding of the geometry of complex geodesics and their invariant left inverses, with implications for holomorphic function theory and complex geometry of non-smooth domains.

Abstract

In the paper we discuss the problem of existence, uniqueness and extension through the boundary of left inverses to complex geodesics in Lempert domains. We concentrate on special left inverses (so called Lempert left inverses) characterized by the fact that their fibers are intersections of affine hyperplanes with the domain.

On left inverses in the Lempert theorem

TL;DR

The paper investigates left inverses to complex geodesics in Lempert domains, introducing a derivative-based construction of Lempert left inverses from given left inverses and establishing a one-parameter family connecting different inverses. It analyzes regularity and boundary-extension obstructions, providing explicit examples in the bidisc and symmetrized bidisc, including royal and flat geodesics, and develops a Burns-Krantz rigidity framework for ellipsoids. The work also discusses the role of dual functions in building left inverses, and addresses invariance under holomorphic automorphisms, highlighting both positive results and notable limitations. Together, these results deepen understanding of the geometry of complex geodesics and their invariant left inverses, with implications for holomorphic function theory and complex geometry of non-smooth domains.

Abstract

In the paper we discuss the problem of existence, uniqueness and extension through the boundary of left inverses to complex geodesics in Lempert domains. We concentrate on special left inverses (so called Lempert left inverses) characterized by the fact that their fibers are intersections of affine hyperplanes with the domain.

Paper Structure

This paper contains 15 sections, 12 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Left inverses induce Lempert left inverses
  3. Regularity of (Lempert) left inverses
  4. Obstruction to regular extensions of left inverses
  5. Examples of domains admitting many Lempert left inverses
  6. The bidisc
  7. The symmetrized bidisc
  8. Lempert left inverse to the royal geodesic
  9. Analysis of the case of flat geodesics
  10. Left inverses
  11. Construction of Lempert left inverses by dual functions
  12. Boundary Schwarz lemma (Burns-Krantz rigidity theorem)
  13. Burns-Krantz rigidity for complex elliposids
  14. Invariance of Lempert left inverses
  15. Introduction

Key Result

Theorem 1.1

Let $D$ be a strongly linearly convex domain in $\mathbb C^n$, $n\geq 2$ or a convex domain in $\mathbb C^n$, $n\geq 1$. Then $c_D\equiv l_D$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 17 more