Harmonic mappings between singular metric spaces

Abstract

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem associated with various energy functionals 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, (1) we extend the existence result of Guo and Wenger (Comm Anal Geom 28(1):89–112, 2020) for solutions to the Dirichlet problem of Korevaar–Schoen energy functional to more general energy functionals in purely singular setting. (2) When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost (Calc Var Partial Differ Equ 5(1):1–19, 1997) and Lin (Analysis on singular spaces, collection of papers on geometry, analysis and mathematical physics, World Science Publishers, River Edge, pp 114–126, 1997) can be adapted to the purely singular setting to yield local Hölder continuity of solutions of the Dirichlet problem of Korevaar–Schoen and Kuwae–Shioya. (3) We extend the Liouville theorem of Sturm (J Reine Angew Math 456:173–196, 1994) for harmonic functions to harmonic mappings between singular metric spaces. (4) We extend the theorem of Mayer (Comm Anal Geom 6:199–253, 1998) 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 \({{\,\mathrm{RCD}\,}}(K,N)\) spaces into NPC spaces. Similar results for the Dirichlet problem associated with the Kuwae–Shioya energy functional and the upper gradient functional are also derived.