An Ancient Stacked Pancake Solution to Mean Curvature Flow

TL;DR

This paper constructs an embedded ancient mean curvature flow in $\mathbb{R}^{n+1}$ for every $n\ge 3$ that qualitatively consists of a stack of two ancient pancakes joined by a catenoidal neck and confined to a fixed slab. The authors introduce a continuity/gluing strategy, building a family of old--but--not--ancient rotationally symmetric approximants and passing to an ancient Brakke flow $B_t$ via Brakke compactness, yielding a global ancient solution for $t\in(-\infty,0]$. The resulting flow $M^n_t$ is $O(n)\times O(1)$-invariant, embedded, non-convex, contained in a slab, and its profile $\Gamma_t$ has exactly two local maxima and one local minimum. This work broadens the catalog of ancient MCF examples trapped in slabs beyond convex cases, informs singularity models and convex-hull properties in spacetime, and hints at possible generalizations to multiple pancakes or halfspace-type configurations.

Abstract

We construct an embedded ancient solution to the mean curvature flow which is qualitatively given by a ``stack'' of two ancient pancakes joined by a neck. Our solution is closed, non-convex, and contained in a slab.

Figures (3)

• Figure 1: A "stack" of two ancient pancakes.
• Figure 2: The profile curve $\Gamma^m_s$.
• Figure 3: Behavior of $(\Sigma^m_s)_{T_m(s)}$ for $m \ll 1$ on the left and for $m \gg 1$ on the right.

Theorems & Definitions (24)

• Theorem 1.1
• Definition 2.1
• Definition 2.2: Brakke flow
• Definition 2.3: Weak set flow
• Definition 2.4: Level set flow
• Theorem 2.1: Theorem 4.1 in aag
• Theorem 2.2: Part of Theorem 1.2 in aag
• Theorem 2.3: Part of Theorem 1.2 in aag
• Theorem 2.4: Theorem 1.1 in aag
• Lemma 2.5