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Gromov-Hausdorff limits and Holomorphic isometries

Claudio Arezzo, Chao Li, Andrea Loi

Abstract

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of Kähler submanifolds of a fixed Kähler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth Kähler manifold satisfying the same curvature bounds, and admitting a holomorphic isometry in the same ambient space. We then apply this convergence result to prove that there are no holomorphic isometries of a non-compact complete Kähler manifold with asymptotically non-negative ones into a finite dimensional complex projective space endowed with the Fubini-Study metric.

Gromov-Hausdorff limits and Holomorphic isometries

Abstract

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of Kähler submanifolds of a fixed Kähler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth Kähler manifold satisfying the same curvature bounds, and admitting a holomorphic isometry in the same ambient space. We then apply this convergence result to prove that there are no holomorphic isometries of a non-compact complete Kähler manifold with asymptotically non-negative ones into a finite dimensional complex projective space endowed with the Fubini-Study metric.
Paper Structure (8 sections, 15 theorems, 70 equations)

This paper contains 8 sections, 15 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Pointed Gromov-Hausdorff convergence
  5. Choices of holomorphic charts
  6. Proof of Theorem \ref{['cptness2']}
  7. Proof of Corollary \ref{['euplane']}
  8. Final remarks

Key Result

Theorem 1.2

Every sequence $\{(M_i,\omega_i,p_i,\phi_i)\} \subset \mathcal{K}(n,R_0,\hat{M},\hat{\omega})$ has a subsequence which converges in the pointed Gromov-Hausdorff sense to a tuple $(M_\infty,\omega_\infty,p_\infty,\phi_\infty) \in \mathcal{K}(n,R_0,\hat{M},\hat{\omega})$.

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 21 more