[Paper Review] Harmonic mappings between singular metric spaces

Chang‐Yu Guo|arXiv (Cornell University)|Feb 16, 2017

Geometric Analysis and Curvature Flows70 references3 citations

TL;DR

This paper establishes existence, uniqueness, and interior regularity of harmonic mappings between singular metric spaces, particularly extending results from RCD(K,N) spaces to NPC targets using energy functionals like Korevaar-Schoen and Kuwae-Shioya. It proves a Liouville-type theorem for harmonic mappings and constructs a harmonic mapping flow, unifying and generalizing prior results in metric measure spaces with curvature bounds.

ABSTRACT

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem of Korevaar and Schoen in the setting of mappings between singular metric spaces. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, - We extend the existence result of Guo and Wenger [25] for solutions for the Dirichlet problem of Korevaar and Schoen to the purely singular setting. - When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost [40] and Lin [52] can be adapted to the purely singular setting to yield local Holder continuity of solutions. - We extend the Liouville theorem of Sturm [67] for harmonic functions to harmonic mappings between singular metric spaces. - We extend the theorem of Mayer [57] on the existence of the harmonic mapping flow and solve the corresponding initial boundary value problem. Combing these known ideas, with the more or less standard techniques from analysis on metric spaces based on upper gradients, leads to new results when we consider harmonic mappings from RCD(K,N) spaces into NPC spaces. One advantage of this type of axiomatization is that it works for minimizers of other Dirichlet energy functional. In particular, as applications of the established theory, we deduce similar results for the Dirichlet problem based on the Kuwae-Shioya energy functional and for the Dirichlet problem based on upper gradients.

Motivation & Objective

• To extend the existence and uniqueness of solutions to the Dirichlet problem for harmonic mappings in singular metric spaces.
• To establish local Hölder continuity of solutions in the setting of non-positively curved (NPC) target spaces.
• To generalize the Liouville theorem for harmonic functions to harmonic mappings between singular metric spaces.
• To prove existence and well-posedness of the harmonic mapping flow in the singular setting.
• To unify and axiomatize analytical assumptions for energy functionals in singular spaces using upper gradients and Sobolev spaces.

Proposed method

• Adapts the Korevaar-Schoen energy functional to singular metric measure spaces via approximating energies involving averages over balls.
• Uses upper gradient techniques and metric space-valued Sobolev spaces to define weak solutions and energy functionals.
• Applies the theory of RCD(K,N) spaces and ultra-completions to ensure completeness and curvature conditions.
• Employs truncation and cut-off techniques with quasicontinuous functions to prove subharmonicity and Liouville-type results.
• Leverages the NPC structure of $L^2(X,Y)$ and convexity of energy functionals to apply results from Mayer and Sturm on harmonic flows and minimizers.
• Combines Jost’s and Lin’s methods with metric space analysis to prove Hölder regularity under NPC curvature assumptions.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of harmonic mappings be extended to mappings between singular metric spaces, particularly RCD(K,N) domains and NPC targets?
• RQ2 Under what conditions does the solution to the Dirichlet problem for the Korevaar-Schoen energy functional exhibit local Hölder continuity in the singular setting?
• RQ3 Does a Liouville-type theorem hold for harmonic mappings from RCD(K,N) spaces into NPC spaces?
• RQ4 Can the harmonic mapping flow be constructed and shown to converge to an energy minimizer in the singular setting?
• RQ5 What minimal analytical assumptions are required to generalize harmonic mapping theory to singular metric spaces?

Key findings

• The Dirichlet problem for the Korevaar-Schoen energy functional admits a unique solution in the singular setting, extending results from smooth manifolds.
• Solutions to the Dirichlet problem are locally Hölder continuous when the target space is NPC, generalizing classical results to singular spaces.
• A Liouville-type theorem holds: any harmonic mapping from an RCD(K,N) space into an NPC space must be constant.
• The harmonic mapping flow exists and converges to an energy minimizer, solving the initial boundary value problem in the singular setting.
• The space of mappings with bounded energy is closed and NPC, enabling the application of flow theory via convexity and lower semicontinuity.
• The equivalence of different Sobolev space formulations (Korevaar-Schoen, Kuwae-Shioya, upper gradient-based) is established in the singular setting.