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arXiv · 1903.12317

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

Hubert Bray·Feng Gui·Zhenhua Liu·Yiyue Zhang
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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.

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Hubert Bray, Feng Gui, Zhenhua Liu, Yiyue Zhang. 2019-03-29. Proof of Bishop's volume comparison theorem using singular soap bubbles. https://arxiv.org/abs/1903.12317

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