A Characterization of the Singular Time of the Mean Curvature Flow

TL;DR

This paper investigates finite-time singularities in mean curvature flow of compact submanifolds, showing they are characterized by the blow-up of a trace involving the second fundamental form.

Contribution

It provides a new characterization of singularities in mean curvature flow through the blow-up of a specific trace related to the second fundamental form.

Findings

Finite-time singularities are characterized by the blow-up of trace A.

The trace A involves the mean curvature and second fundamental form.

This characterization helps understand the nature of singularities in mean curvature flow.

Abstract

In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.