Surfaces of bounded mean curvature in Riemannian manifolds

TL;DR

This paper proves a compactness result for sequences of bounded mean curvature surfaces in Riemannian manifolds, showing convergence of maps and metrics, and deriving diameter bounds based on geometric constraints.

Contribution

It establishes a new compactness theorem for surfaces with bounded mean curvature, extending Gromov's approach to this geometric setting.

Findings

Convergence of inclusion maps in $C^0$ topology.

Pullback metrics converge to a fractal pseudo-metric.

Diameter bounds depend on curvature, area, and genus constraints.

Abstract

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in $C^0$ to a map from a surface of genus $g$ to $M$. We also show that, on passing to a further subsequence, the distance functions corresponding to pullback metrics converge to a pseudo-metric of fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\subset M$ together with bounds on the geometry of $M$ give an upper bound on the diameter of $F$. Our proof is modelled on Gromov's compactness theorem for $J$-holomorphic curves.