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Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

Mat Langford, James McCoy

TL;DR

This work delivers local curvature-pinching estimates for fully nonlinear one-homogeneous curvature flows in general ambient spaces by adapting the localized Huisken--Stampacchia framework. It unifiedly treats umbilic, convexity, and cylindrical estimates for convex, concave, inverse-concave, and Lynch interpolated speeds, including flows in curved ambient geometries. Central to the approach are carefully designed cutoffs, $G_{\varepsilon,\sigma}$-type pinching quantities, and a Stampacchia bootstrap that converts $L^2$ control into $L^\infty$ bounds, enabling precise localization of curvature pinching. The results extend classical estimates from mean curvature flow to a broad class of fully nonlinear flows, providing a versatile toolkit for analyzing singularities and long-time behavior in curved spaces. The framework also accommodates Lynch's speeds and outlines how the localization persists in Riemannian ambient spaces, highlighting the practical impact for geometric analysis and applications requiring curvature pinching control under general speeds.

Abstract

In a recent article, a localization of the Huisken--Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.

Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

TL;DR

This work delivers local curvature-pinching estimates for fully nonlinear one-homogeneous curvature flows in general ambient spaces by adapting the localized Huisken--Stampacchia framework. It unifiedly treats umbilic, convexity, and cylindrical estimates for convex, concave, inverse-concave, and Lynch interpolated speeds, including flows in curved ambient geometries. Central to the approach are carefully designed cutoffs, -type pinching quantities, and a Stampacchia bootstrap that converts control into bounds, enabling precise localization of curvature pinching. The results extend classical estimates from mean curvature flow to a broad class of fully nonlinear flows, providing a versatile toolkit for analyzing singularities and long-time behavior in curved spaces. The framework also accommodates Lynch's speeds and outlines how the localization persists in Riemannian ambient spaces, highlighting the practical impact for geometric analysis and applications requiring curvature pinching control under general speeds.

Abstract

In a recent article, a localization of the Huisken--Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.
Paper Structure (21 sections, 17 theorems, 115 equations)

This paper contains 21 sections, 17 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curvature functions and admissible flow speeds
  4. Cut-off functions
  5. A Poincaré-type inequality
  6. The Sobolev inequality of Michael and Simon
  7. Stampacchia's lemma
  8. Hypersurfaces evolving by convex speeds
  9. Proof of the convexity estimate for flows of hypersurfaces by convex admissible speeds
  10. Proof of the cylindrical (and umbilic) estimates for flows of hypersurfaces by convex admissible speeds
  11. Surfaces evolving by general speeds
  12. Proof of the umbilic estimate for flows of surfaces by general admissible speeds
  13. Proof of the convexity estimate for flows of surfaces by general admissible speeds
  14. Hypersurfaces evolving by concave speeds
  15. Proof of the cylindrical estimates for flows by concave speeds
  16. ...and 6 more sections

Key Result

Proposition 3

Let $\Gamma^n \subset \mathbb{R}^n$ be an open, symmetric cone satisfying $\overline{\Gamma}{}^n\backslash \left\{ 0 \right\} \subset \left\{w \in \mathbb{R}^n: w_1+\dots+w_n>0\right\}$ and $\overline{\Gamma}{}^n\cap \mathrm{Cyl} = \emptyset$. There is a constant $\gamma = \gamma(n,\Gamma^n) > 0$ wi Here $H$ denotes the mean curvature.

Theorems & Definitions (20)

  • Proposition 3: Poincaré-type inequality MR3669776
  • Proposition 4: Michael--Simon Sobolev inequality MR0344978
  • Proposition 5: Poincaré inequality
  • Lemma 6: Stampacchia's lemma MR0251373MR1786735
  • Theorem 7: Convexity estimate --- convex speeds
  • Theorem 8: Cylindrical estimates --- convex speeds
  • Lemma 9
  • proof
  • Lemma 10: $L^2$-estimate
  • Theorem 11: Umbilic estimate --- surfaces
  • ...and 10 more