Ancient mean curvature flows with finite total curvature

TL;DR

This paper constructs a family of ancient mean curvature flows over minimal hypersurfaces with finite total curvature, demonstrating their convergence, finite total curvature, and mean convexity in some cases.

Contribution

It introduces a new family of ancient flows over minimal hypersurfaces with finite total curvature, showing their convergence and geometric properties.

Findings

Flows have finite total curvature and finite mass drop.

Existence of a mean convex flow within the family.

Exponential convergence in space-time variables.

Abstract

We construct an $I$-family of ancient graphical mean curvature flows over a minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the Morse index $I$ by establishing exponentially fast convergence in terms of $|x|^2-t$. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.