Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity

2026/03/31 by Mario Santilli
Mathematics · #math.AP #math.DG

paper · pdf

v2: introduction revised

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Given a sequence of uniformly convex norms ϕ_h on Rⁿ⁺¹ converging to an arbitrary norm ϕ, we prove rigidity of L¹ -accumulation points of sequences of sets E_h ⊆ Rⁿ⁺¹ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with ϕ_h . Here, almost criticality is measured in terms of the Lⁿ -deviation from being constant of the distributional anisotropic mean ϕ_h -curvature of (the varifold associated to) of the reduced boundaries of E_h . We prove that such limits are finite union of disjoint, but possibly mutually tangent, ϕ-Wulff shapes.

Citations

• A sharp quantitative Alexandrov inequality and applications to volume preserving geometric flows in 3D
• Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with L¹-type deviations
• Rigidity and compactness with constant mean curvature in warped product manifolds
• Curvature measures and soap bubbles beyond convexity
• On volume-preserving crystalline mean curvature flow
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• On the shape of compact hypersurfaces with almost constant mean curvature
• Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures