Calculus of Variations and Geometric Measure Theory

M. Vedovato

Quantitative regularity for $p$-minimizing maps through a Reifenberg Theorem

created by vedovato on 07 Oct 2019
modified on 01 Jan 2023


Published Paper

Inserted: 7 oct 2019
Last Updated: 1 jan 2023

Journal: The Journal of Geometric Analysis
Year: 2021

ArXiv: 1910.01971 PDF
Links: Link to published article


In this article we extend to generic p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done by Cheeger and Naber in 2013. Then, adapting the work of Naber and Valtorta, we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is $k$-rectifiable for a k which only depends on $p$ and the dimension of the domain.

Keywords: harmonic maps, quantitative stratification, reifenberg theorem