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Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces. (English) Zbl 1041.30020

Let $\left(M,\sigma \right)$ and $\left(N,\rho \right)$ be hyperbolic Riemann surfaces of finite analytic type $\left(g,n\right)$, $f:M\to N$ a quasiconformal harmonic diffeomorphism and $\varphi \left(z\right)\phantom{\rule{0.166667em}{0ex}}d{z}^{2}$ the Hopf differential of $f$. Two inequalities are proved: ${\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}_{M}|\varphi |\le A\left(g,n\right){e}^{2d\left(\tau ,{\tau }_{0}\right)}$, and $\parallel \text{Hopf}\left(f\right)\parallel \le C\left(g,n\right){e}^{2d\left(\tau ,{\tau }_{0}\right)}$, where $\tau \in$ Teich$\left(M\right)$ is the point in the Teichmüller space represented by the pair $\left(N,f\right)$, ${\tau }_{0}$ is the origin in this space and $d$ is the Teichmüller distance. The constants $A\left(g,n\right)=$ Area$\left(N,\rho \right)$ and $C\left(g,n\right)$ depend only on the type $\left(g,n\right)$.

The second estimate is applied to the deformation theory of Kleinian groups: two quasiconformal related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. A partial answer is given to the conjecture of R. Schoen about existence of harmonic quasiconformal maps with prescribed boundary values. The author proves that every symmetric map of the unit circle can be extended to a unique harmonic quasiconformal map of the unit disk, and every quasisymmetric map whose boundary dilatation is less than $P$ can be extended to a unique harmonic quasiconformal map of the unit disk, where $P$ is a universal constant from L. F. Tam and T. Wan [J. Differ. Geom. 42, No 2, 368–410 (1995; Zbl 0873.32019)].

##### MSC:
 30F60 Teichmüller theory 30C65 Quasiconformal mappings in ${ℝ}^{n}$ and other generalizations 32G15 Moduli of Riemann surfaces, Teichmüller theory