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Planarity and convexity for pinched ancient solutions of mean curvature flow

Tang-Kai Lee, Keaton Naff, Jingze Zhu

TL;DR

This work advances the understanding of ancient mean curvature flows in higher codimension under uniform pinching by establishing a parabolically scale-invariant planarity estimate and a codimension-one convexity result in the hypersurface setting. The planarity analysis combines a pointwise differential inequality with a Gaussian-weighted integral approach, yielding sharp control on the nonprincipal second fundamental form $\hat{A}$ in terms of the mean curvature $H$ and time, while the convexity result uses a weighted energy method to force the smallest principal curvature to be nonnegative. Together these estimates imply rigidity and classification outcomes for pinched ancient flows and shrinkers, showing that under finite entropy and polynomial growth, high-codimension flows must in fact be codimension one and convex, with shrinkers falling into generalized-cylinder types. The paper also discusses extensions to noncompact settings and outlines open questions about optimal pinching constants, potential surgeries, and the removal of curvature-growth assumptions, signaling several directions for future research in high-codimensional mean curvature flow.

Abstract

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle--Huisken--Sinestrari \cite{BHS}. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.

Planarity and convexity for pinched ancient solutions of mean curvature flow

TL;DR

This work advances the understanding of ancient mean curvature flows in higher codimension under uniform pinching by establishing a parabolically scale-invariant planarity estimate and a codimension-one convexity result in the hypersurface setting. The planarity analysis combines a pointwise differential inequality with a Gaussian-weighted integral approach, yielding sharp control on the nonprincipal second fundamental form in terms of the mean curvature and time, while the convexity result uses a weighted energy method to force the smallest principal curvature to be nonnegative. Together these estimates imply rigidity and classification outcomes for pinched ancient flows and shrinkers, showing that under finite entropy and polynomial growth, high-codimension flows must in fact be codimension one and convex, with shrinkers falling into generalized-cylinder types. The paper also discusses extensions to noncompact settings and outlines open questions about optimal pinching constants, potential surgeries, and the removal of curvature-growth assumptions, signaling several directions for future research in high-codimensional mean curvature flow.

Abstract

We prove a parabolically scale-invariant variation of the planarity estimate in \cite{Na22} for higher codimension mean curvature flow, borrowing ideas from work of Brendle--Huisken--Sinestrari \cite{BHS}. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.

Paper Structure

This paper contains 14 sections, 15 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Planarity estimate for ancient solutions
  3. Convexity estimate for ancient solutions
  4. Consequences of the main estimates
  5. Planarity estimate
  6. Pointwise estimate
  7. Integral estimate
  8. Convexity estimate
  9. Consequences of planarity and convexity estimates
  10. Some open questions on higher codimension $c\,$-pinched MCF
  11. A gradient estimate in non-convex regions
  12. Dimensional constants that arise in the planarity estimate
  13. The constant $\frac{4}{3n}$ and the preservation of $c$-pinching
  14. The constant $\frac{3(n+1)}{2n(n+2)}$ and the planarity estimate

Key Result

Theorem 1.1

Suppose $n \geq 2$ and $N > n + 1$. Let $M^n_t = F(M^n, t) \subset \mathbb{R}^N$, $t\in[0,T]$, be a smooth family of $n$-dimensional, complete, connected, immersed submanifolds evolving by MCF. Suppose each time slice $M_t$ has bounded curvature. Finally, suppose there exists a constant $\epsilon_0 holds pointwise on $M \times (0, T]$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 2.1
  • Lemma 2.4
  • proof
  • ...and 21 more