Quantitative regularity for p-minimizing maps through a Reifenberg Theorem

2019/10/04 by Vedovato, Mattia
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.01971

Abstract

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.

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