Deformation of contracting maps under the harmonic map heat flow

TL;DR

This paper investigates how contracting maps between positively curved closed manifolds deform under the harmonic map heat flow, emphasizing the impact of curvature pinching and singular value conditions.

Contribution

It extends the understanding of the rigidity of contracting maps by analyzing their deformation under harmonic map heat flow with new curvature and singular value conditions.

Findings

Identifies conditions under which contracting maps remain rigid during deformation.

Establishes a relationship between curvature pinching and singular value constraints.

Provides new insights into the deformation behavior of maps in positively curved manifolds.

Abstract

In this paper, inspired by the work Lee-Wan, we researched the rigidity of contracting maps between closed manifolds with positive curvature. We focused on the relation between curvature pinching and contracting conditions involving arbitrary $k$ singular values.