Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of area non-increasing maps

Man-Chun Lee, Luen-Fai Tam, Jingbo Wan

Abstract

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of $\mathbb{CP}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature.

Rigidity of area non-increasing maps

Abstract

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of , is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive -isotropic curvature.
Paper Structure (9 sections, 15 theorems, 135 equations)

This paper contains 9 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: graphical mean curvature flow, short time and long time existence
  3. Mean curvature flow equation and short-time existence
  4. Geometry of graph and area non-increasing maps
  5. Long-time existence
  6. Proof of the Main Theorems
  7. Evolution equations under Uhlenbeck trick
  8. Monotonicity in static background
  9. Monotonicity in evolving background

Key Result

Theorem 1.1

Let $(M^m, g), (N^n,h)$ be two compact manifolds with $m, n\ge 3$. Suppose $f_0$ is a smooth map from $M$ to $N$ which is area non-increasing. Suppose one of the following curvature conditions is satisfied: Here $\kappa_M, \kappa_N$ are the lower bounds of the sectional curvature of $M, N$ respectively, $\tau_N$ is the upper bound of the sectional curvature of $N$, $\text{\rm Ric}^g_{\min}$ is th

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.1
  • proof
  • Claim 2.1
  • Lemma 3.1
  • ...and 21 more