Canonical metrics on complex manifold

Abstract

Complex manifolds are topological spaces that are covered by coordinate charts where the coordinate changes are given by holomorphic transformations. For example, Riemann surfaces are one dimensional complex manifolds. In order to understand complex manifolds, it is useful to introduce metrics that are compatible with the complex structure. In general, we should have a pair (M, ds 2 M ) where ds 2 M is the metric. The metric is said to be canonical if any biholomorphisms of the complex manifolds are automatically isometries. Such metrics can naturally be used to describe invariants of the complex structures of the manifold.The first important examples of such metrics were constructed by Poincaré for Riemann surfaces with genus greater than one. Note that the flat metric on the torus is not quite canonical unless we require the biholomorphisms to preserve the area. (In higher dimensions, this is the same as preserving the Kähler class.) Poincaré's metrics are metrics with constant negative curvature. It requires a proof of the existence theorem for conformal deformation of metrics. It is a nonlinear differential equation and hence a difficult higher dimensional problem. Generalizations of Poincaré's work took some time.Besides Riemannian metrics that are compatible with the complex structure, there are other types of metrics which we shall discuss in the following. Carathéodory metricThis is a metric on a Riemann surface X constructed by the following procedure. Given a tangent vector v at a point p ∈ X, the Carathéodory metric is defined by v = sup{ f * (v) : f is a holomorphic map from X to the unit disk equipped with the Poincaré metric}.In the 1930s, Ahlfors proved a powerful generalization of the Schwarz lemma in complex analysis. Theorem 1. Holomorphic maps from the Poincaré disk to a Riemann surface with curvature less than −1 is distance decreasing.Grauert-Reckziegel generalized Ahlfors-Schwarz lemma in the 1960s for maps from disk into higher dimensional complex hermitian manifold with strongly negative curvature. In 1978, I [1]