Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harmonic mappings between singular metric spaces

Chang-Yu Guo

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.

Harmonic mappings between singular metric spaces

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.

Paper Structure

This paper contains 29 sections, 29 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness
  3. Interior regularity
  4. Liouville theorem
  5. Harmonic mapping flow
  6. Preliminaries
  7. Metric spaces of non-positive curvature
  8. Strong measure contraction properties
  9. Ultra-completions of metric spaces
  10. Metric space valued Sobolev spaces
  11. Korevaar-Schoen Sobolev spaces
  12. Kuwae-Shioya Sobolev spaces
  13. Sobolev spaces based on upper gradients
  14. Equivalence of different Sobolev spaces
  15. Cheeger energy and $\mathop{\mathrm{RCD}}\nolimits(K,N)$ spaces
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be a compact metric space and $Y$ a metric space that is 1-complemented in some ultra-completion of $Y$. Suppose $X$ admits an energy functional $\mathcal{E}$ with property ${\mathcal{B}}$. Fix a domain $\Omega\subset X$. Then, for each $\phi\in S^{1,2}(X,Y)$, there exists a mapping

Theorems & Definitions (78)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9: Existence and boundedness of gradient flow of the Dirichlet Energy
  • Corollary 1.10
  • ...and 68 more