Regularity of Harmonic Maps from Polyhedra to CAT(1) Spaces

2016/10/25 by Christine Breiner, Ailana Fraser, Breiner, Christine +9
Mathematics · #53C43 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1610.07829

openalex publication_date 2016/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n-2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^(k) with k ≤ n-2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

Citations

• Quasisymmetric parametrizations of two-dimensional metric spheres

Related

• Lipschitz continuity of harmonic maps between \rm RCD(K,N) spaces and \rm CAT(κ) spaces
• Optimal boundary regularity of harmonic maps from RCD(K,N)-spaces to CAT(0)-spaces
• Hölder continuity of energy minimizer maps between riemannian polyhedra
• Comments on the regularity of harmonic maps between singular spaces
• Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) spaces