An isoperimetric constant associated to horizons in S3S3 blown up at two points

Let gg be a metric on S3S3 with positive Yamabe constant. When blowing up gg 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 ΘΘ-invariant for gg which is an isoperimetric constant for the cylindrical domain inside the outermost minimal surface of the blown-up metric. Further we find relations between ΘΘ and the Yamabe constant and the existence of horizons in the blown-up metric on R3R3.