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Generic regularity of isoperimetric regions in dimension eight

Kobe Marshall-Stevens, Gongping Niu, Davide Parise

TL;DR

This work proves that in closed 8-dimensional Riemannian manifolds, isoperimetric regions admit smooth, nondegenerate boundaries for a generic choice of metric and enclosed volume (and also for fixed volumes with generic metrics). The authors develop a global perturbation strategy built around the twisted Jacobi operator, introduce semi-nondegeneracy and a singular-capacity framework to quantify and remove isolated singularities, and apply a Sard–Smale type argument in pseudo-neighborhoods together with a cone decomposition for almost minimisers. A key achievement is showing that the singular capacity can be iteratively reduced to zero in the generic setting, thereby forcing smooth boundaries for all isoperimetric regions under generic metric-volume data. The results extend quantitative isoperimetric inequalities to dimension eight and lay groundwork for broader applications in geometric analysis of constant mean curvature boundaries and generic regularity phenomena.

Abstract

We establish generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight. In particular, we show that every isoperimetric region has a smooth nondegenerate boundary for a generic choice of smooth metric and enclosed volume, or for a fixed enclosed volume and a generic choice of smooth metric.

Generic regularity of isoperimetric regions in dimension eight

TL;DR

This work proves that in closed 8-dimensional Riemannian manifolds, isoperimetric regions admit smooth, nondegenerate boundaries for a generic choice of metric and enclosed volume (and also for fixed volumes with generic metrics). The authors develop a global perturbation strategy built around the twisted Jacobi operator, introduce semi-nondegeneracy and a singular-capacity framework to quantify and remove isolated singularities, and apply a Sard–Smale type argument in pseudo-neighborhoods together with a cone decomposition for almost minimisers. A key achievement is showing that the singular capacity can be iteratively reduced to zero in the generic setting, thereby forcing smooth boundaries for all isoperimetric regions under generic metric-volume data. The results extend quantitative isoperimetric inequalities to dimension eight and lay groundwork for broader applications in geometric analysis of constant mean curvature boundaries and generic regularity phenomena.

Abstract

We establish generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight. In particular, we show that every isoperimetric region has a smooth nondegenerate boundary for a generic choice of smooth metric and enclosed volume, or for a fixed enclosed volume and a generic choice of smooth metric.

Paper Structure

This paper contains 24 sections, 33 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Structure
  4. Notation, isoperimetric regions, and twisted Jacobi fields
  5. Notation
  6. Preliminaries on isoperimetric regions
  7. Twisted Jacobi fields and a spectral theorem
  8. Asymptotic rates and metric perturbations
  9. Definitions and perturbation functions
  10. Induced twisted Jacobi fields and metric perturbations
  11. Bumpy metric volume pairs
  12. Pseudo-neighbourhoods and three compactness lemmas
  13. A Sard--Smale theorem for metric volume pairs
  14. Generic semi-nondegeneracy
  15. Singular capacity for isoperimetric regions
  16. ...and 9 more sections

Key Result

Theorem 1

In a closed manifold of dimension eight, every isoperimetric region has smooth nondegenerate boundary for a generic choice of smooth Riemannian metric and enclosed volume.

Theorems & Definitions (103)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Corollary 1
  • Lemma 1: Compactness of isoperimetric regions
  • proof
  • Remark 2
  • Lemma 2: Fixed volume mean curvature bound
  • proof
  • Remark 3
  • ...and 93 more