A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria

TL;DR

This paper introduces a new monotone quantity in mean curvature flow for graphs of maps between Riemannian manifolds, leading to sharp homotopic criteria for maps between complex projective spaces and spheres.

Contribution

It identifies a novel monotone quantity in graphical mean curvature flows of higher codimensions, establishing sharp homotopic criteria for certain classes of maps.

Findings

Monotone quantity increases under area-decreasing conditions.

Flow induces a natural homotopy of the map.

Sharp criteria for homotopy classes of maps between complex projective spaces and spheres.

Abstract

A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.