On Ecker's local integral quantity at infinity for ancient mean curvature flows

TL;DR

This paper shows that Ecker's local integral quantity and Huisken's global integral quantity are equivalent at infinity for ancient mean curvature flows, linking their finiteness and implications for entropy.

Contribution

It establishes the equivalence of Ecker's local integral and Huisken's global integral at infinity under finiteness conditions, connecting local and global geometric quantities.

Findings

Ecker's local integral equals Huisken's global integral at infinity.

Finiteness of Ecker's integral implies finiteness of entropy at infinity.

The result applies to ancient mean curvature flows with finite Huisken's integral on each time-slice.

Abstract

We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancient mean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity.