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Optimal regularity for minimizers of the prescribed mean curvature functional over isotopies

Lorenzo Sarnataro, Douglas Stryker

TL;DR

This work develops a sharp regularity theory for minimizers of the $h$-prescribed mean curvature functional $\mathcal{A}^h$ over isotopy classes, achieving optimal $C^{1,1}$ regularity by extending the AS and MSY strategies to the $\mathcal{A}^h$ setting. It introduces a robust machinery—including fill-ins, resolution of overlaps, replacement, and the $n$-membrane framework—to control topology, obtain strong compactness, and upgrade regularity via free boundary techniques (WZ_multiple). A central outcome is a min-max existence result in the round $S^3$ showing that, for an open dense set of prescribing functions with $\|h\|_{L^{\infty}}\le 0.547$, there exists an embedded sphere with prescribed mean curvature; the method yields explicit density/mass controls and provides sharp regularity up to $C^{1,1}$. The results thus deliver a quantitative, topologically controlled existence theory for prescribed mean curvature spheres in $S^3$, with potential extensions to more general metrics and curvature regimes. Key innovations include the precise $C^{1,1}$ regularity theory, the gamma-reduction paradigm for isotopy minimization, and a min-max framework that tightly couples area, volume, and mean curvature through controlled variational and geometric-analytic tools.

Abstract

We prove the optimal $C^{1,1}$ regularity for minimizers of the prescribed mean curvature functional over isotopy classes. As an application, we find an embedded sphere of prescribed mean curvature in the round 3-sphere for an open dense set of prescribing functions with $L^{\infty}$ norm at most 0.547.

Optimal regularity for minimizers of the prescribed mean curvature functional over isotopies

TL;DR

This work develops a sharp regularity theory for minimizers of the -prescribed mean curvature functional over isotopy classes, achieving optimal regularity by extending the AS and MSY strategies to the setting. It introduces a robust machinery—including fill-ins, resolution of overlaps, replacement, and the -membrane framework—to control topology, obtain strong compactness, and upgrade regularity via free boundary techniques (WZ_multiple). A central outcome is a min-max existence result in the round showing that, for an open dense set of prescribing functions with , there exists an embedded sphere with prescribed mean curvature; the method yields explicit density/mass controls and provides sharp regularity up to . The results thus deliver a quantitative, topologically controlled existence theory for prescribed mean curvature spheres in , with potential extensions to more general metrics and curvature regimes. Key innovations include the precise regularity theory, the gamma-reduction paradigm for isotopy minimization, and a min-max framework that tightly couples area, volume, and mean curvature through controlled variational and geometric-analytic tools.

Abstract

We prove the optimal regularity for minimizers of the prescribed mean curvature functional over isotopy classes. As an application, we find an embedded sphere of prescribed mean curvature in the round 3-sphere for an open dense set of prescribing functions with norm at most 0.547.
Paper Structure (56 sections, 41 theorems, 317 equations, 4 figures)

This paper contains 56 sections, 41 theorems, 317 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Application to existence in the 3-sphere
  3. Idea of the proof of regularity
  4. Idea of the proof of existence in the 3-sphere
  5. Outline of the paper
  6. Acknowlegements
  7. Notation and Conventions
  8. Examples
  9. Failure of prescribed mean curvature
  10. Failure of smoothness
  11. Minimization over disks
  12. Preliminaries
  13. Local control
  14. Thin tube isoperimetric inequality
  15. Area comparison
  16. ...and 41 more sections

Key Result

Theorem 1.1

Let $h: M \to \mathbb{R}$ be a smooth function, and let $c := \sup_M |h|$. Let $U \subset M$ be a sufficiently small (see §sec:MSYintreg for a precise definition) open set with $C^1$ boundary, and let $\Omega_0 \subset M$ be an open subset with smooth boundary having tranverse intersection with $\pa for $\varepsilon_k \to 0$. Then there is a varifold $V$, an open set $\Omega \subset U$, and a subs

Figures (4)

  • Figure 1: The left image depicts the initial stack of two disks. The middle image depicts a replacement for the bottom disk that intersects the top disk. The right image depicts a possible disentanglement of the disks.
  • Figure 2: $\varepsilon = 0.4$
  • Figure 3: $\varepsilon = 2-\sqrt{3}$
  • Figure 4: $\varepsilon = 0.1$

Theorems & Definitions (94)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.5
  • Theorem 1.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Lemma 4.2: MSY
  • ...and 84 more