Ancient mean curvature flows from minimal hypersurfaces

TL;DR

The paper constructs an $I$-parameter family of embedded ancient mean curvature flows in $\mathbb{R}^{n+1}$ emanating from an unstable minimal hypersurface $\Sigma^n$ with finite total curvature, where $I$ is the Morse index of the Jacobi operator. It linearizes the graphical MCF about $\Sigma$ and uses the finite negative spectrum of the stability operator $L=\Delta_\Sigma+|A_\Sigma|^2$ to drive a finite-dimensional modulation, while controlling the remaining directions via weighted parabolic theory. A global heat-kernel analysis, weighted Schauder estimates, and nonlinear error bounds lead to a contraction mapping in a weighted function space, producing an $I$-parameter family of ancient solutions that converge to $\Sigma$ exponentially as $t\to -\infty$. This work extends the landscape of non-self-similar ancient solutions and highlights how unstable minimal hypersurfaces govern backward-in-time MCF dynamics with finite total curvature.

Abstract

For $n\geq 2$, we construct $I$-dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface $Σ$ with finite total curvature in $\mathbb{R}^{n+1}$, where $I$ is the Morse index of the Jacobi operator on $Σ$.

Theorems & Definitions (24)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof