Proof of Bishop's volume comparison theorem using singular soap bubbles

TL;DR

This paper offers a novel proof of Bishop's volume comparison theorem by employing isoperimetric hypersurfaces, or 'soap bubbles,' addressing the challenge posed by their potential singularities.

Contribution

The paper introduces a new proof technique for Bishop's theorem using singular soap bubbles, expanding the methods available for geometric comparison theorems.

Findings

Proof using isoperimetric hypersurfaces with singularities

Overcoming challenges posed by codimension 7 singularities

Alternative approach to classical volume comparison proof

Abstract

Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome.