Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result

TL;DR

This paper introduces an abstract regularity principle that improves the estimation of the Minkowski dimension of higher multiplicity points in Dir-minimizing Q-valued functions, with broader applications to related problems.

Contribution

It develops a general regularity framework that refines dimension estimates for singular sets in Dir-minimizing functions and extends to other mathematical contexts.

Findings

New estimate of the Minkowski dimension of higher multiplicity points

Recovery and improvement of existing results in the literature

Applicability of the abstract principle to various problems

Abstract

In this note we prove an abstract version of a recent quantitative stratifcation priciple introduced by Cheeger and Naber (Invent. Math., 191 (2013), no. 2, 321-339; Comm. Pure Appl. Math., 66 (2013), no. 6, 965-990). Using this general regularity result paired with an $\varepsilon$-regularity theorem we provide a new estimate of the Minkowski dimension of the set of higher multiplicity points of a Dir-minimizing Q-valued function. The abstract priciple is applicable to several other problems: we recover recent results in the literature and we obtain also some improvements in more classical contexts.