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.
Categories: math.DG math.GT