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A geometric flow on noncompact affine Riemannian manifolds

Heming Jiao, Hanzhang Yin

Abstract

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature and bounded geometry is diffeomorphic to $\mathbb{R}^n$ if its tangent bundle has maximal volume growth.

A geometric flow on noncompact affine Riemannian manifolds

Abstract

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature and bounded geometry is diffeomorphic to if its tangent bundle has maximal volume growth.
Paper Structure (6 sections, 33 theorems, 171 equations)

This paper contains 6 sections, 33 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Short time existence
  4. A priori estimates for the geometric flow
  5. existence criteria for the geometric flow
  6. Uniformization

Key Result

Theorem 1.1

Let $(M, \nabla, g_0)$ be a complete noncompact affine Riemannian manifold. Assume that (i) (ii) Then $S_A=S_B$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 46 more