Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

Abstract

We study the smallest area $A(M,g)$ of a 2-dimensional stationary integral varifold in a closed Einstein 4-manifold $(M^4,g)$ with $Ric_g = \lambda g, |\lambda|\leq 3, Vol(M,g)\geq v>0, diam(M,g)\leq D, H_1(M;\mathbb{Z})=0.$ Building on the previous work on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class, there is an upper bound $A(M,g)\leq F_{Ein}(v,D),$ where $F_{Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\varepsilon$-regularity constants for Einstein metrics.