Harmonic maps and framed $\mathrm{PSL}_2(\mathbb{C})$-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 $\mathrm{PSL}_2(\mathbb{C})$-representation $\hatρ = (ρ,β)$, there is a $ρ$-equivariant harmonic map $f:\mathbb{H}^2 \to \mathbb{H}^3$ that is asymptotic to the framing $β$. 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.