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Connectedness properties of small minimal clusters in Riemannian or Finsler manifolds

Stefano Nardulli, Aldo Pratelli

TL;DR

This work addresses the small-volume behavior of isoperimetric clusters on compact manifolds endowed with Riemannian or Finsler structures. It develops a local-to-global framework by establishing diameter control for tiny components and a fixed-norm reduction, enabling precise statements about connectedness: at most $m$ components in general Finsler manifolds and full connectedness in the Riemannian case. The authors then derive asymptotic expansions for the multi-isoperimetric profile, showing it matches the Euclidean model at first order (with explicit constants in the Finsler setting) and validating a tangent-space-based perspective for small-volume clusters. These results unify and extend classical isoperimetric theory to the multi-chamber setting, with potential applications to phase transition models and existence/multiplicity questions in geometric variational problems.

Abstract

We prove that in a compact Riemannian manifold, the $m$-minimal clusters of sufficiently small total volume are connected and with small diameter, while in a more general Finsler manifold they are done by at most $m$ connected components of small diameter. We apply these results to calculate the asymptotic expansion of the multi-isoperimetric profile at the first nontrivial order, for small volumes.

Connectedness properties of small minimal clusters in Riemannian or Finsler manifolds

TL;DR

This work addresses the small-volume behavior of isoperimetric clusters on compact manifolds endowed with Riemannian or Finsler structures. It develops a local-to-global framework by establishing diameter control for tiny components and a fixed-norm reduction, enabling precise statements about connectedness: at most components in general Finsler manifolds and full connectedness in the Riemannian case. The authors then derive asymptotic expansions for the multi-isoperimetric profile, showing it matches the Euclidean model at first order (with explicit constants in the Finsler setting) and validating a tangent-space-based perspective for small-volume clusters. These results unify and extend classical isoperimetric theory to the multi-chamber setting, with potential applications to phase transition models and existence/multiplicity questions in geometric variational problems.

Abstract

We prove that in a compact Riemannian manifold, the -minimal clusters of sufficiently small total volume are connected and with small diameter, while in a more general Finsler manifold they are done by at most connected components of small diameter. We apply these results to calculate the asymptotic expansion of the multi-isoperimetric profile at the first nontrivial order, for small volumes.
Paper Structure (7 sections, 9 theorems, 217 equations)

This paper contains 7 sections, 9 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Notation and basic definitions
  3. The small connected components
  4. The general Finsler case: proof of Theorem A
  5. The Riemannian case: proof of Theorem B
  6. Asymptotic Equivalence of the Isoperimetric Profiles and final remarks
  7. Final remarks

Key Result

Theorem 1.1

Let $M$ be a $N$-dimensional, ${\rm C}^2$, compact Riemannian manifold, and let $m\in\mathbb N$ be given. Then where $J_M$ and $J_{\mathbb R^N}$ denote the multi-isoperimetric profiles of $M$ and of $\mathbb R^N$ with the Euclidean metric, and for any vector $v\in(\mathbb R^+)^m$ we denote by $|v|=v_1 + v_2 + \cdots + v_m$ the $L^1$ norm, which is the "total volume" of an $m$-cluster whose volume

Theorems & Definitions (29)

  • Theorem 1.1: Asymptotic expansion of the multi-isoperimetric profile for small volumes
  • Definition 1.2: Fixed-norm manifold
  • Definition 1.3: oscillation of an atlas
  • Definition 1.4: maximal diameter
  • Definition 1.5: $\sigma$-minimal deviation
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 19 more