Quantitative estimate on singularities in isoperimetric clusters

TL;DR

This paper provides a quantitative estimate on the number of specific singularities in almost minimizing clusters, with implications for the finiteness of boundary classes in volume-constrained minimizers and applications to higher-dimensional minimal surfaces.

Contribution

It introduces a general method to estimate and bound the number of certain singularities in isoperimetric clusters and related minimal surface problems.

Findings

Bound on the number of triple junctions in 2D clusters.

Bound on the number of tetrahedral points in 3D clusters.

Finite classification of boundary types for volume-constrained minimizers.

Abstract

We prove a quantitative estimate on the number of certain singularities in almost minimizing clusters. In particular, we consider the singular points belonging to the lowest stratum of the Federer-Almgren stratification (namely, where each tangent cone does not split a $\R$) with maximal density. As a consequence we obtain an estimate on the number of triple junctions in $2$-dimensional clusters and on the number of tetrahedral points in $3$ dimensions, that in turn implies that the boundaries of volume-constrained minimizing clusters form at most a finite number of equivalence classes modulo homeomorphism of the boundary, provided that the prescribed volumes vary in a compact set. The method is quite general and applies also to other problems: for instance, to count the number of singularities in a codimension 1 area-minimizing surface in $\R^8$.