Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow

TL;DR

This paper establishes convergence relations between varifolds, flat chains, and rectifiable currents, providing insights into singularities in mean curvature flow.

Contribution

It proves that convergence of integral varifolds leads to convergence of associated flat chains and currents, revealing new links between these geometric objects.

Findings

Convergence of integral varifolds implies convergence of mod 2 flat chains.

Subsequential convergence of integer-multiplicity rectifiable currents is established.

Results impose restrictions on singularities in mean curvature flow.

Abstract

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.