An upper bound for the smallest area of a minimal surface in manifolds of dimension four

TL;DR

This paper establishes an upper bound on the minimal surface area in certain 4D manifolds with specific geometric constraints, linking it to the manifold's homological filling function.

Contribution

It provides a new bound on the smallest surface area in 4D manifolds based on homological filling functions and geometric bounds, extending understanding of minimal surfaces in higher dimensions.

Findings

Bound depends only on volume and diameter

Uses homological filling function estimation

Applicable to manifolds with trivial first homology

Abstract

In this paper, we prove that for any closed 4-dimensional Riemannian manifold $M$ with trivial first homology group, if the Ricci curvature $|Ric|\leq3$, the diameter $diam(M)\leq D$ and the volume $vol(M)>v>0$, then the area of a smallest 2-dimensional stationary integral varifold in $M$ is bounded by F(v,D), for some function F that only depends on v and D. Our bound for the area is based on the estimation of the first homological filling function of $M$.