Extremal Hermitian metrics on Riemannian surfaces. (English) Zbl 0955.30032

Chen, Xiuxiong

Int. Math. Res. Not. 1998, No. 15, 781-797 (1998).

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.

Reviewer: Stancho Dimiev (Sofia)

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

Keywords:

Riemann surface with boundary; conformal metric; compact Riemann; extremal Hermitian metric