2019/03/29 by Bray, Hubert, Gui, Feng, Liu, Zhenhua +1 · 1 citation
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.12317
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.