Harmonic maps in singular geometry and rigidity

TL;DR

The survey analyzes harmonic maps into NPC/CAT$(0)$ spaces, including targets without smooth structure, and shows how regularity and convexity properties yield rigidity phenomena for group actions and geometric structures. It surveys the classical smooth-target theory (Eells–Sampson, Bochner) and extends to NPC targets via Korevaar–Schoen, culminating in Gromov–Schoen rigidity results and their generalizations to DM-complexes, hyperbolic/hyperbolic-buildings, and Teichmüller space. Key innovations include the DM-complex framework, regularity theorems with codimension-2 singular sets, asymptotic product structures, and monotonicity tools adapted to non-smooth targets, enabling both finite-energy and pluriharmonic rigidity results, including holomorphic rigidity in Teichmüller settings. The work also discusses infinite-energy pluriharmonic maps and highlights ongoing challenges and potential directions for extending harmonic-map rigidity to broader non-smooth geometries and group actions.

Abstract

This survey reviews results on harmonic maps into spaces of non-positive curvature, with a focus on targets that lack smooth structure. More precisely, we consider targets that are complete metric spaces with non-positive curvature in the sense of Alexandrov, commonly referred to as NPC (non-positively curved) or CAT(0) spaces. We discuss applications of harmonic maps to rigidity phenomena, including generalizations of Margulis superrigidity and the holomorphic rigidity of Teichmüller space. Our approach relies heavily on the regularity theory of harmonic maps to non-smooth targets, enabling differential-geometric techniques to be employed in the absence of any smooth structure on the target.

Figures (8)

• Figure 1: Triangle comparison
• Figure 2: tripod
• Figure 3: $u_0$, $u_1$ and interpolation map $u_t$
• Figure 4: Codimension 1 singular set
• Figure 5: Projection of vertical leaves

Theorems & Definitions (25)

• Theorem 1
• Theorem 3: korevaar-schoen Theorem 2.2 and 2.4.6
• Definition 4
• Theorem 5
• Remark 6
• Theorem 7: Uniqueness in CAT($\kappa$) with $\kappa<0$ meseUniqueness
• Theorem 8: Uniqueness in Euclidean buildings daskal-meseMRL
• Theorem 9: Rank 1 $p$-adic superrigidity
• Definition 10
• Definition 11