Communications in Analysis and Geometry

Volume 27 (2019)

Number 6

Quantitative estimate on singularities in isoperimetric clusters

Pages: 1233 – 1249



Maria Colombo (Institute for Theoretical Studies, Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland; and Institut für Mathematik, Universität Zürich, Switzerland)

Luca Spolaor (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


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 $\mathbb{R}^8$.

Received 27 September 2016

Accepted 13 June 2017

Published 12 December 2019