Harmonic maps for Hitchin representations
Qiongling Li

TL;DR
This paper investigates harmonic maps associated with Hitchin representations, establishing energy bounds and conditions for equality, and demonstrating that such maps are distance-increasing immersions with unique cases of equality.
Contribution
It proves energy density bounds for harmonic maps from hyperbolic surfaces to symmetric spaces and characterizes when equality occurs, linking to Fuchsian representations.
Findings
Energy density of harmonic maps is at least 1.
Equality in energy density occurs only for base n-Fuchsian representations.
Hitchin representations induce distance-increasing minimal immersions.
Abstract
Let be a hyperbolic surface, be a Hitchin representation for , and be the unique -equivariant harmonic map from to the corresponding symmetric space. We show its energy density satisfies and equality holds at one point only if and is the base -Fuchsian representation of . In particular, we show given a Hitchin representation for , every -equivariant minimal immersion from a hyperbolic plane into the corresponding symmetric space is distance-increasing, i.e. . Equality holds at one point only if it holds everywhere and is an -Fuchsian representation.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric and Algebraic Topology · Geometry and complex manifolds
