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Diameter, Area, and Mean curvature

Gregory R. Chambers, Jared Marx-Kuo

TL;DR

The paper develops intrinsic diameter bounds for submanifolds $M^m\hookrightarrow N^n$ with boundary and ambient manifolds under curvature constraints, extending classical results of Simon, Topping, Wu–Zheng. It leverages the Hoffman–Spruck Michael–Sobolev inequality with quantities $M(x,R)$ and $\kappa(x,R)$, together with a Vitali-type ball covering to relate mean curvature, volume, and geodesic length, yielding bounds of the form $d_{int}(M) \leq C(k_0,r_0,m)\bigl( \int_M |H|^{m-1}\,d\mu + \mathrm{Vol}(M) \bigr)$, and a boundary version with an extra $\mathrm{Vol}(M)^{1/m}$ term plus a boundary term. The results interpolate between earlier diameter estimates and apply to PMC/CMC surfaces arising in min-max constructions, including extensions to non-smooth metrics. Consequently, diameter control is achieved under broader ambient and boundary conditions, enabling finite-diameter conclusions for a wider class of geometric variational problems.

Abstract

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean curvature and the area of the manifold. We apply our diameter bounds to minimal, constant mean curvature, and prescribed mean curvature surfaces arising in min-max constructions.

Diameter, Area, and Mean curvature

TL;DR

The paper develops intrinsic diameter bounds for submanifolds with boundary and ambient manifolds under curvature constraints, extending classical results of Simon, Topping, Wu–Zheng. It leverages the Hoffman–Spruck Michael–Sobolev inequality with quantities and , together with a Vitali-type ball covering to relate mean curvature, volume, and geodesic length, yielding bounds of the form , and a boundary version with an extra term plus a boundary term. The results interpolate between earlier diameter estimates and apply to PMC/CMC surfaces arising in min-max constructions, including extensions to non-smooth metrics. Consequently, diameter control is achieved under broader ambient and boundary conditions, enabling finite-diameter conclusions for a wider class of geometric variational problems.

Abstract

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean curvature and the area of the manifold. We apply our diameter bounds to minimal, constant mean curvature, and prescribed mean curvature surfaces arising in min-max constructions.

Paper Structure

This paper contains 5 sections, 12 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and Lemmas
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of Theorem \ref{['thm:boundary']}
  5. Applications

Key Result

Theorem 1

Suppose that $M$ is an $m$-dimensional Riemannian manifold without boundary, $N$ is an $n$-dimensional Riemannian manifold, and $M \rightarrow N$ is an isometric immersion. We make the following assumptions on $N$: Then the intrinsic diameter of $M$, denoted by $d_{int}(M)$, satisfies where $H$ is the mean curvature vector of $M$ in $N$ and $\overline{V}_{0} = \overline{V}_0(m, r_0, k_0) > 0$ is

Figures (1)

  • Figure 1: Visualization of one ball argument

Theorems & Definitions (22)

  • Theorem 1
  • Remark
  • Theorem 2
  • Remark
  • Corollary 1
  • Corollary 2
  • Theorem 3: Thm 2.1 HS
  • Definition 1
  • Definition 2
  • Lemma 4
  • ...and 12 more