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

Abstract

In this paper, we prove that for any closed 4-dimensional Riemannian manifold $M$ with trivial first homology group, if the Ricci curvature $|Ric|\le 3$, the diameter $diam(M)\le D$ 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 $M$.

References in corpus (1)

• Regularity of Einstein Manifolds and the Codimension 4 Conjecture