An Upper Bound for the Smallest Area of a Minimal Surface in Manifolds of Dimension Four

Abstract

In this paper, we prove that if M is a closed 4-dimensional Riemannian manifold with trivial first homology group, Ricci curvature $$|Ric|\le 3$$, diameter $${{\,\mathrm{diam}\,}}(M)\le D$$, and volume $${{\,\mathrm{vol}\,}}(M)>v>0$$, then the smallest area of a 2-dimensional minimal surface in M is bounded by F(v, D), for some function F that only depends on v and D. In order to prove this result, we first establish upper bounds for the first homological filling function of M that are of independent interest. This part of our work is based on recent results of Cheeger and Naber about manifolds with Ricci curvature bounded from both sides.