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Extremal Hermitian metrics on Riemannian surfaces. (English) Zbl 0955.30032
The author develops the program started in his PhD dissertation, namely the study of the variational problem of minimizing the energy functional $$E(g)$$ in the variational space of all conformal metrics $$g$$ with finite energy and area on a compact Riemann surface. The motivation is related with a possible generalization of the classical uniformization theory to surfaces with boundary. Some positive results are announced. The most impressive is the one for a Dirichlet boundary value problem, especially about the existence of extremal Hermitian metrics with prescribed length and geodesic curvature along the boundary. The author’s approach is, in fact, to show that there exists an extremal Hermitian metric in each conformal class as it is described above. In this geometric context the obtained PDE is more appropriate to study according the understanding of the author. The proofs of some main theorems are outlined.

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 30F20 Classification theory of Riemann surfaces 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30E25 Boundary value problems in the complex plane 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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