An isoperimetric constant associated to horizons in $S^3$ blown-up at two points

TL;DR

This paper introduces a new isoperimetric constant called the $ heta$-invariant for metrics on $S^3$ with positive Yamabe constant, relating it to horizons and minimal surfaces after a blow-up process at two points.

Contribution

It defines the $ heta$-invariant for blown-up metrics on $S^3$ and explores its relation to the Yamabe constant and horizon existence in scalar flat manifolds.

Findings

$ heta$-invariant provides bounds related to minimal surfaces

Relations established between $ heta$ and Yamabe constant

Conditions for horizons in blown-up metrics on $ ^3$

Abstract

Let $g$ be a metric on $S^3$ with positive Yamabe constant. When blowing up $g$ at two points, a scalar flat manifold with two asymptotically flat ends is produced and this manifold will have compact minimal surfaces. We introduce the $\Th$-invariant for $g$ which is an isoperimetric constant for the cylindrical domain inside the outermost minimal surface of the blown-up metric. Further we find relations between $\Th$ and the Yamabe constant and the existence of horizons in the blown-up metric on $\mR^3$.