Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

TL;DR

This paper establishes an upper bound on the minimal area of certain 2-dimensional varifolds in closed Einstein 4-manifolds, linking geometric properties to homological filling functions and regularity constants.

Contribution

It introduces a bound on the minimal area of stationary varifolds in Einstein 4-manifolds based on homological filling functions and regularity constants, extending previous work.

Findings

Bound on minimal area $A(M,g)$ depending only on volume and diameter

Connection between homological filling functions and Einstein metric regularity

Applicable to Einstein 4-manifolds with specified Ricci curvature and topology

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.