Mean curvature flows and isotopy problems

TL;DR

This paper explores the behavior of mean curvature flows of graph maps between Riemannian manifolds, providing global existence results and applications to isotopy problems in geometry and topology.

Contribution

It offers new estimates for the flow as a non-linear parabolic system and presents several global existence theorems with applications to geometric and topological isotopy problems.

Findings

Establishment of global existence theorems for mean curvature flows

Development of estimates for the flow as a non-linear parabolic system

Applications to isotopy problems in geometry and topology

Abstract

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.