Let $\bar{X}$ be a compact Riemann surface of genus $g$, and let $D:=\{p_{1},p_{2},\cdots,p_{n}\}$ be $n$ distinct points on $\bar{X}$. Define $X:=\bar{X}-D$ to be the punctured Riemann surface given by the complement of the divisor $D$. Assume that $2g-2+n>0$, which ensures that the surface $X$ supports a complete hyperbolic metric $h_{0}$.
Let $\sigma$ and $\rho$ be two smooth hyperbolic metrics on the $C^{\infty}$ surface $\bar{X}$. It is known that there exists a unique harmonic diffeomorphism from $(\bar{X},\sigma)$ to $(\bar{X}, \rho)$ in the isotropy class of the identity map $\rm{id}\colon\bar{X}\rightarrow \bar{X}$. This follows from work of Eells and Sampson, Hartman, Schoen and Yau. Moreover, Lohkamp extended this unique existence result to a punctured surface. That is, given a complete hyperbolic metric $h$ on the $C^{\infty}$ surface $X$, there is a unique diffeomorphism $f$, of $X$ homotopic to the identity map, such that $f$ is a harmonic map from $(X,h_{0})$ to $(X,h)$. An alternative proof is provided by Biswas, Ares-Gastesi and Govindarajan using parabolic Higgs bundle ([1], p.g. 1557). Below is the proof.
Proof. Let $(V, \nabla)$ be the flat rank two vector bundle given by the Fuchsian representation for the Riemann surface $(X,h)$. Let $H$ be the harmonic metric on $V$ given by the main theorem of [2] (p.g. 755 ) for the flat bundle $(V, \nabla)$ on the Riemann surface $(X,h_{0})$. In other words, $H$ gives a section, denoted by $s$, of the associated fiber bundle with fiber ${\rm SL}(2,\mathbb{R})/{\rm SO}(2)=\mathbb{H}$, where $\mathbb{H}$ is the upper half plane. This section $s$ gives the harmonic map $f$ mentioned above.
Question: How can we obtain the harmonic map $f$ from the section $s$ ?
I. Biswas, P. Ares-Gastesi and S. Govindarajan, Parabolic Higgs bundles and Teichmuller spaces for punctured surfaces. Trans. Amer. Math. Soc. 163 (1972), 261--275.
C. Simpson, Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3 (1990), 713--770.
