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Open problems on compact constant mean curvature surfaces with boundary

Rafael López

TL;DR

This article aggregates easily stated open problems in the theory of compact cmc surfaces with boundary in $\mathbb{R}^3$ and surveys the current state of knowledge. It emphasizes circle-boundary problems, including conjectures that planar disks and spherical caps are the only compact embedded, disk-like, and stable cmc surfaces with boundary $\mathbb{S}^1$, and discusses related methods such as the Alexandrov reflection principle and rigidity via the Hopf differential. It then analyzes arbitrary boundary curves through the flux formula, clarifying how the mean curvature $H$ is constrained by boundary data and outlining existence questions for a range of $H$ values and boundary configurations. Finally, it reviews the Dirichlet problem for the cmc equation, summarizing solvability criteria under convexity and curvature hypotheses, known barrier-based estimates, and several open questions about convexity and graphicality of cmc graphs on convex domains.

Abstract

We present a collection of easily stated open problems in the theory of compact constant mean curvature surfaces with boundary. We also survey the current status of answering them.

Open problems on compact constant mean curvature surfaces with boundary

TL;DR

This article aggregates easily stated open problems in the theory of compact cmc surfaces with boundary in and surveys the current state of knowledge. It emphasizes circle-boundary problems, including conjectures that planar disks and spherical caps are the only compact embedded, disk-like, and stable cmc surfaces with boundary , and discusses related methods such as the Alexandrov reflection principle and rigidity via the Hopf differential. It then analyzes arbitrary boundary curves through the flux formula, clarifying how the mean curvature is constrained by boundary data and outlining existence questions for a range of values and boundary configurations. Finally, it reviews the Dirichlet problem for the cmc equation, summarizing solvability criteria under convexity and curvature hypotheses, known barrier-based estimates, and several open questions about convexity and graphicality of cmc graphs on convex domains.

Abstract

We present a collection of easily stated open problems in the theory of compact constant mean curvature surfaces with boundary. We also survey the current status of answering them.
Paper Structure (4 sections, 1 theorem, 10 equations, 2 figures)

This paper contains 4 sections, 1 theorem, 10 equations, 2 figures.

Table of Contents

  1. Introduction and preliminaries
  2. Part I: circular boundary
  3. Part II: arbitrary boundary curve
  4. The Dirichlet problem for the cmc equation

Key Result

Theorem 1.1

Figures (2)

  • Figure 1: Surfaces with constant mean curvature spanning a circle $\mathbb S^1$ obtained by intersecting a sphere with a plane (left). The middle surfaces are (small and big) spherical caps; the right surface is a planar disk.
  • Figure 2: Left: a compact embedded surface over the boundary plane $P$. Right: an embedded surface where the Alexandrov reflection method fails at $p$. Notice that the surface $\Sigma$ is star-shaped from the point $q$.

Theorems & Definitions (1)

  • Theorem 1.1