A note on sigular time of mean curvature flow

TL;DR

This paper demonstrates that under specific geometric conditions, the mean curvature flow of a compact submanifold does not develop singularities at infinite time within certain Riemannian manifolds.

Contribution

It establishes conditions preventing singularity formation at infinite time for mean curvature flow in manifolds with bounded geometry and controlled curvature and volume growth.

Findings

Mean curvature flow avoids singularities at infinity under specified conditions.

Bounded geometry and curvature constraints are sufficient to prevent infinite-time singularities.

Provides theoretical insights into long-term behavior of mean curvature flow.

Abstract

We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions .