Lipschitz regularity of harmonic map heat flows from RCD spaces into CAT(0) spaces

2026/08/04 by Bang-Xian Han, Hui-Chun Zhang, Xi-Ping Zhu
Mathematics · #math.MG #math.AP #math.DG #msc:53C43 #msc:58E20 #msc:31E05 #msc:35K55 #msc:49Q22

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arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

Harmonic map heat flows into complete CAT(0) spaces admit a variational construction even when the target is singular, but their regularity is less understood when the source is also nonsmooth. We prove positive-time local Lipschitz regularity for the EVI gradient flow of the Korevaar--Schoen energy from a finite-dimensional RCD(K,N) space into a complete CAT(0) target, under a bounded-image assumption on the initial data. The resulting representative is locally Lipschitz jointly in space and time. The proof combines a metric Hamilton--Jacobi argument with an elliptic contact estimate applied on suitable time slices of the RCD source.

Citations

• ABP estimate on metric measure spaces via optimal transport
• On the regularity of harmonic maps from \sf RCD(K,N) to \sf CAT(0) spaces and related results
• Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD(K,N) spaces to CAT(0) spaces
• Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds
• Korevaar-Schoen's energy on strongly rectifiable spaces
• Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows
• Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows
• Heat Kernel Bounds on Metric Measure Spaces and Some Applications
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• The Li–Yau inequality and heat kernels on metric measure spaces
• Lipschitz continuity of harmonic maps between Alexandrov spaces
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,∞) metric measure spaces
• On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• On the differential structure of metric measure spaces and applications
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Duality on gradient estimates and Wasserstein controls
• Metric Spaces of Non-Positive Curvature
• Harmonic maps into singular spaces and p -adic superrigidity for lattices in groups of rank one
• Harmonic Mappings of Riemannian Manifolds