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

We show that given an element $X$ of the enhanced Teichmüller space ${\mathcal{T}}^{\pm }(\mathbb{S},\mathbb{M})$ and a type-preserving framed ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$-representation $\widehat{\rho }=(\rho ,\beta )$, 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.