[Paper Review] Character varieties and harmonic maps to R-trees

Georgios Daskalopoulos, Stamatis Dostoglou+1 more|ArXiv.org|Oct 6, 1998

Geometric and Algebraic Topology13 references4 citations

TL;DR

This paper establishes that the Korevaar-Schoen limit of equivariant harmonic maps associated with a sequence of irreducible SL₂(ℂ) representations of a compact Riemannian manifold's fundamental group converges to an equivariant harmonic map into an ℝ-tree. The key result shows that the length function of this limiting map is projectively equivalent to the Morgan-Shalen limit of the representations, linking character varieties to harmonic maps into metric trees.

ABSTRACT

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.

Motivation & Objective

• To understand the geometric limit of equivariant harmonic maps associated with sequences of irreducible SL₂(ℂ) representations of a compact Riemannian manifold’s fundamental group.
• To analyze the structure of the limiting object when such harmonic maps converge, particularly in relation to R-trees.
• To establish a correspondence between the Morgan-Shalen limit of character varieties and the length function of the limiting harmonic map into an ℝ-tree.
• To investigate the implications of fixed ends under the group action on the limiting ℝ-tree.

Proposed method

• Utilizes the Korevaar-Schoen theory of harmonic maps into metric spaces, particularly ℝ-trees, as a framework for taking limits of harmonic maps.
• Applies the concept of equivariant harmonic maps from the universal cover of a compact Riemannian manifold to metric spaces, preserving group actions.
• Employs the Morgan-Shalen compactification of character varieties to analyze the asymptotic behavior of representations.
• Shows that the length function of the limiting harmonic map into an ℝ-tree is projectively equivalent to the Morgan-Shalen limit of the sequence of representations.
• Analyzes the action of the fundamental group on the limiting ℝ-tree, particularly when the action fixes an end.
• Uses minimal harmonic maps into ℝ-trees as a tool to relate geometric and algebraic limits in representation theory.

Experimental results

Research questions

• RQ1 What is the nature of the limit of a sequence of equivariant harmonic maps associated with irreducible SL₂(ℂ) representations of a compact manifold’s fundamental group?
• RQ2 How does the length function of the limiting harmonic map into an ℝ-tree relate to the Morgan-Shalen limit of the character variety?
• RQ3 Under what conditions does the group action on the limiting ℝ-tree fix an end, and what are the geometric consequences?
• RQ4 Can the Korevaar-Schoen limit of harmonic maps be characterized as a minimal harmonic map into an ℝ-tree?
• RQ5 Is there a projective equivalence between the length function of the limiting harmonic map and the Morgan-Shalen limit of the representation sequence?

Key findings

• The Korevaar-Schoen limit of a sequence of equivariant harmonic maps to symmetric spaces is an equivariant harmonic map into an ℝ-tree.
• The limiting harmonic map into the ℝ-tree is minimal, meaning it achieves the infimum of energy in its homotopy class.
• The length function associated with the limiting harmonic map is projectively equivalent to the Morgan-Shalen limit of the sequence of representations.
• When the group action on the limiting ℝ-tree fixes an end, the harmonic map exhibits specific dynamical and geometric constraints.
• The construction provides a geometric realization of the Morgan-Shalen compactification via harmonic map theory.
• The result establishes a deep link between representation theory, harmonic maps, and metric geometry through the lens of ℝ-trees.