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

Abstract

We show that given an element $X$ of the enhanced Teichmüller space ${\mathcal{T}}^{\pm }(\mathbb{S},\mathbb{M})$ and a type-preserving framed ${\mathrm{PSL}}_2(\mathbb{C})$-representation $\hat{\ddot{I}}=(\ddot{I},{\hat{I}}^2)$, there is a $\ddot{I}$-equivariant harmonic map $f:{\mathbb{H}}^2\to {\mathbb{H}}^3$ that is asymptotic to the framing ${\hat{I}}^2$. 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.

27 pages, 3 figures