Isoperimetric Regions in Anisotropically Scaled Product Manifolds

2025/12/10 by Efstratios Vernadakis, Vernadakis, Efstratios
Mathematics · #Geometric Analysis and Curvature Flows #Analytic and geometric function theory #Morphological variations and asymmetry

paper · pdf · doi:10.48550/arxiv.2512.09490

Abstract

Let M, N be compact Riemannian manifolds. Then, for fixed volume fraction, in the product of a sufficiently small homothetic copy of M with N, every isoperimetric region is the product of M with an isoperimetric region in N, provided the boundaries of the isoperimetric regions in N are smooth.

Citations

• The Riemannian Quantitative Isoperimetric Inequality
• Steiner and Schwarz symmetrization in warped products and fiber bundles with density
• In polytopes, small balls about some vertex minimize perimeter

Related

• Existence of singular isoperimetric regions
• Isoperimetric Regions in Nonpositively Curved Manifolds
• Isoperimetric inequalities in cylinders with density
• Soap bubbles and isoperimetric regions in the product of a closed manifold with Euclidean space
• The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field