Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

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 $Ri{c}_g=\lambda g,\mathrm{\mid }\lambda \mathrm{\mid }\le 3,Vol(M,g)\ge v>0,diam(M,g)\le D,H_1(M;\mathbb{Z})=0.$ 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 $A(M,g)\le F_{Ein}(v,D),$ where $F_{Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\epsilon$-regularity constants for Einstein metrics.