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Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$

Da Rong Cheng

TL;DR

The paper proves the existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$ confined by a convex barrier, by constructing a Sacks–Uhlenbeck type perturbation $E_{\varepsilon,p,f}$ and applying a min-max scheme to obtain critical points with Morse index at most $1$ for almost every $H\in(0,H_0)$. It develops a robust variational framework with a volume term $V_f$ and a local reduction to handle the path dependence, then passes to the limit $\varepsilon\to 0$ to produce smooth free boundary cmc disks; under convexity of the barrier, it achieves the full-range existence for $H\in(0,H_0)$ by combining an index comparison with a uniform energy bound via the Hersch trick. The approach yields a tight index control across the limiting process, enabling continuation across the measure-zero set of $H$ values and providing a convex-barrier improvement of Struwe's classical results. Overall, the work extends Struwe’s strategy beyond minimal disks, offering a systematic variational route to free boundary cmc disks in $\mathbb{R}^3$ with explicit barrier structure and quantitative energy/index estimates.

Abstract

Given a surface $Σ$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $Σ$, there exists a branched immersed disk which has constant mean curvature $H$ and boundary meeting $Σ$ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on $Σ$. Specifically, when $Σ$ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H \in (0, H_0)$. Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of $H$'s is a Morse index upper bound.

Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$

TL;DR

The paper proves the existence of free boundary disks with constant mean curvature in confined by a convex barrier, by constructing a Sacks–Uhlenbeck type perturbation and applying a min-max scheme to obtain critical points with Morse index at most for almost every . It develops a robust variational framework with a volume term and a local reduction to handle the path dependence, then passes to the limit to produce smooth free boundary cmc disks; under convexity of the barrier, it achieves the full-range existence for by combining an index comparison with a uniform energy bound via the Hersch trick. The approach yields a tight index control across the limiting process, enabling continuation across the measure-zero set of values and providing a convex-barrier improvement of Struwe's classical results. Overall, the work extends Struwe’s strategy beyond minimal disks, offering a systematic variational route to free boundary cmc disks in with explicit barrier structure and quantitative energy/index estimates.

Abstract

Given a surface in diffeomorphic to , Struwe (Acta Math., 1988) proved that for almost every below the mean curvature of the smallest sphere enclosing , there exists a branched immersed disk which has constant mean curvature and boundary meeting orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on . Specifically, when itself is convex and has mean curvature bounded below by , we obtain existence for all . Instead of the heat flow used by Struwe, we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou (arXiv:2012.13379), a key ingredient for extending existence across the measure zero set of 's is a Morse index upper bound.
Paper Structure (16 sections, 27 theorems, 260 equations)

This paper contains 16 sections, 27 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. The perturbed functional
  3. Function spaces and special coordinates
  4. The enclosed volume
  5. The perturbed functional
  6. First variation and a Palais-Smale condition
  7. Estimates for critical points of the perturbed functional
  8. Smoothness of critical points
  9. A priori estimates for critical points
  10. A maximum principle
  11. Second variation and index
  12. Existence of critical points of the perturbed functional
  13. Passage to the limit as $\varepsilon \to 0$
  14. Improved existence result under convexity assumptions
  15. An index comparison result
  16. ...and 1 more sections

Key Result

Theorem 1.1

For almost every $H \in (0, H_0)$, there exists a non-constant, free boundary, branched immersion $u: (\overline{\mathbf{B}}, \partial \mathbf{B}) \to (\overline{\Omega'}, \Sigma)$ with constant mean curvature $H$ and Morse index at most $1$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 45 more