Diameter, Area, and Mean curvature

TL;DR

This paper extends diameter bounds for submanifolds with boundary in ambient manifolds with curvature restrictions, linking diameter to mean curvature integrals and area, with applications to various mean curvature surfaces.

Contribution

It generalizes existing diameter bounds to include submanifolds with boundary and non-compact ambient spaces under mild curvature conditions.

Findings

Diameter bounds depend on mean curvature integrals and area.

Applications to minimal and constant mean curvature surfaces.

Provides new tools for min-max surface constructions.

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.