Harmonic maps and framed $\mathrm{PSL}_2(\mathbb{C})$-representations

TL;DR

This paper establishes the existence and uniqueness of certain harmonic maps from punctured Riemann surfaces to hyperbolic 3-space, associated with framed PSL(2,C)-representations, using harmonic map heat flow techniques.

Contribution

It proves the existence and uniqueness of $ ho$-equivariant harmonic maps asymptotic to framings for framed PSL(2,C)-representations, linking harmonic maps with enhanced Teichmüller space.

Findings

Existence of harmonic maps asymptotic to framings.

Uniqueness of harmonic maps given principal parts of Hopf differentials.

Application of harmonic map heat flow in the proof.

Abstract

We show that given an element $X$ of the enhanced Teichm\"{u}ller space $\mathcal{T}^\pm(\mathbb{S}, \mathbb{M})$ and a type-preserving framed $\mathrm{PSL}_2(\mathbb{C})$-representation $\hat{\rho} = (\rho,\beta)$, there is a $\rho$-equivariant harmonic map $f:\mathbb{H}^2 \to \mathbb{H}^3$ that is asymptotic to the framing $\beta$. Here, the domain is the universal cover of the punctured Riemann surface obtained from a conformal completion of $X$. Moreover, such a harmonic map is unique if one prescribes, in addition, the principal part of the Hopf differential at each puncture. The proof uses the harmonic map heat flow.