Generic regularity and nondegeneracy of isoperimetric regions in higher dimensions

Establish that, for a generic smooth Riemannian metric, isoperimetric regions in higher-dimensional closed manifolds have smooth boundaries and nondegenerate constrained (twisted) Jacobi operators, including in the presence of general singular sets.

Background

The paper explains that singularities of isoperimetric boundaries complicate deformation theory because standard Sard–Smale arguments based on smooth normal-graph parametrizations do not directly handle perturbations across singular points. Previous work established generic smoothness and nondegeneracy for generic metric–volume pairs in dimension eight, where the singular set can consist of finitely many isolated points, using compactness of the Jacobi form domain derived from nonconcentration estimates near those singularities.

For isoperimetric boundaries with more general singular sets, the required compactness argument has not been extended. Consequently, the paper identifies generic boundary regularity and generic nondegeneracy of the constrained Jacobi operator as unresolved problems in higher dimensions. The present paper addresses generic uniqueness instead and does not resolve these regularity and nondegeneracy questions.