Calculus of Variations and Geometric Measure Theory

M. Colombo - L. Spolaor

Quantitative estimate on singularities in isoperimetric clusters

created by colombom on 16 Jan 2017
modified by spolaor on 05 Oct 2017


Accepted Paper

Inserted: 16 jan 2017
Last Updated: 5 oct 2017

Journal: CAG
Year: 2017


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 $\mathbb R$) 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 $\mathbb R^8$.