Ancient mean curvature flows with finite total curvature

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.

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