*(English)*Zbl 1200.58010

The Teichmüller space for an open manifold was introduced by the second author in [Asian J. Math. 2, No. 2, 355–404 (1998; Zbl 1045.58006)] and was studied in detail for Riemann surfaces.

In the paper under review, the authors investigate harmonic diffeomorphisms between complete Riemann surfaces of negative curvature. More precisely, besides many other results, they prove the following:

Let $({M}^{2},{g}_{0})$ and $({M}^{2},{g}_{1})$ be two open Riemann surfaces. Let ${\left\{{g}_{t}\right\}}_{0\le t\le 1}$ be a curve in the Sobolov topology, ${K}_{{g}_{t}}\equiv -1$, $inf\sigma \left({{\Delta}}_{0}\left({g}_{t}\right)\right)>0$, ${r}_{\text{inj}}\left({g}_{t}\right)>0$, $0\le t\le 1$. Then there exists a unique harmonic diffeomorphism ${f}_{1}:({M}^{2},{g}_{0})\to ({M}^{2},{g}_{1})$ which is isotopic by harmonic diffeomorphisms to $\text{id}:({M}^{2},{g}_{0})\to ({M}^{2},{g}_{0})$ in the unit component ${\mathcal{D}}_{0}^{r+1}$ of the completed diffeomorphism group ${\mathcal{D}}^{r+1}$.

The technique essentially relies on the framework of non-linear Sobolov analysis on open manifolds as developed by the second author in [Global analysis on open manifolds. New York, NY: Nova Science Publishers (2007; Zbl 1188.58001)]. Application to Teichmüller theory for open surfaces is also given.