Chen, Qun; Eichhorn, Jürgen Harmonic diffeomorphisms between complete Riemann surfaces of negative curvature. (English) Zbl 1200.58010 Asian J. Math. 13, No. 4, 473-534 (2009). 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 \(\{ g_t\}_{0 \leq t \leq 1}\) be a curve in the Sobolov topology, \(K_{g_t} \equiv -1\), \(\inf \sigma(\Delta_0(g_t)) >0\), \(r_{\text{inj}}(g_t) >0\), \(0 \leq t \leq 1\). 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. Reviewer: Mahender Singh (Chennai) Cited in 2 Documents MSC: 58D27 Moduli problems for differential geometric structures 58E20 Harmonic maps, etc. 30F60 Teichmüller theory for Riemann surfaces Keywords:harmonic map; open surfaces; Teichmüller space; diffeomorphism Citations:Zbl 1045.58006; Zbl 1188.58001 PDFBibTeX XMLCite \textit{Q. Chen} and \textit{J. Eichhorn}, Asian J. Math. 13, No. 4, 473--534 (2009; Zbl 1200.58010) Full Text: DOI Euclid