Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hermitian metric on hyperbolic complex manifolds

Debraj Chakrabarti, Prachi Mahajan

TL;DR

The paper introduces κ_M, an invariant Hermitian metric on Kobayashi hyperbolic manifolds constructed by applying the complex Binet-Legendre inner product to the Kobayashi-Busemann metric. This approach mirrors Wu’s idea of building a nearly distance-decreasing Hermitian metric from the Kobayashi data but emphasizes regularity, with κ_M matching the Poincaré-Bergman metric on the unit ball and preserving smoothness properties from hat{k}_M. The authors establish key properties: biholomorphic invariance, continuity and completeness under Kobayashi completeness, and explicit distortion bounds under holomorphic maps; they also prove compatibility with products and coverings. Comparisons with the Wu metric yield distance-equivalence results, suggesting κ_M as a viable alternative tool for analyzing hyperbolic complex geometry and curvature questions, with potential implications for Kobayashi’s conjectures. The work combines Finsler-to-Hermitian transition techniques with complex analysis invariants to produce a metric that is both theoretically appealing and amenable to curvature studies.

Abstract

We describe a method of defining a Hermitian metric on Kobayashi hyperbolic manifolds. The metric is distance decreasing under holomorphic mappings, up to a multiplicative constant. This method is distinct from the classical construction of Wu, and yields a metric which is expected to have superior regularity properties.

A Hermitian metric on hyperbolic complex manifolds

TL;DR

The paper introduces κ_M, an invariant Hermitian metric on Kobayashi hyperbolic manifolds constructed by applying the complex Binet-Legendre inner product to the Kobayashi-Busemann metric. This approach mirrors Wu’s idea of building a nearly distance-decreasing Hermitian metric from the Kobayashi data but emphasizes regularity, with κ_M matching the Poincaré-Bergman metric on the unit ball and preserving smoothness properties from hat{k}_M. The authors establish key properties: biholomorphic invariance, continuity and completeness under Kobayashi completeness, and explicit distortion bounds under holomorphic maps; they also prove compatibility with products and coverings. Comparisons with the Wu metric yield distance-equivalence results, suggesting κ_M as a viable alternative tool for analyzing hyperbolic complex geometry and curvature questions, with potential implications for Kobayashi’s conjectures. The work combines Finsler-to-Hermitian transition techniques with complex analysis invariants to produce a metric that is both theoretically appealing and amenable to curvature studies.

Abstract

We describe a method of defining a Hermitian metric on Kobayashi hyperbolic manifolds. The metric is distance decreasing under holomorphic mappings, up to a multiplicative constant. This method is distinct from the classical construction of Wu, and yields a metric which is expected to have superior regularity properties.

Paper Structure

This paper contains 14 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. From norms to inner products
  3. The real Binet-Legendre construction
  4. The complex BL construction
  5. Basic properties
  6. The metric $\kappa_M$
  7. The Kobayashi-Royden pseudometric
  8. The Kobayashi-Busemann metric and the definition of $\kappa_M$
  9. Comparison with the Wu metric
  10. Proofs of the Theorems
  11. Proof of Theorem \ref{['thm-main']}
  12. Proof of Theorem \ref{['thm-product']}
  13. Proof of Theorem \ref{['thm-covering']}
  14. Some questions

Key Result

Theorem 1.1

On every Kobayashi hyperbolic complex manifold $M$, there is defined an invariant Hermitian metric $\kappa_M$ which has the following properties:

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.3
  • Proposition 2.5
  • proof
  • Proposition 2.10
  • proof
  • Definition 3.3