Holomorphic Maps and Conformal Transformation on Hermitian Manifolds

Abstract

Let (N,h) be a Hermitian manifold and let φ: N → IR+ be a C∞ map, then \( (\rm N,\tilde h\, = \,\phi ^4 h) \) also is a Hermitian manifold. If h is a Kähler metric, the metric \( \rm \tilde h\, = \,\phi ^4 h \) in general is not a Käh1er metric. But under certain conditions we can obtain some properties of (N,h) by using conformal transformation. In this paper we optain the following three results: 1) the Schwarz lemmas of ho1omorphic maps between complex manifolds whose curvatures are bounded from above by nonnegative constants; 2) Liouville-type theorems of ho1omorphic maps; 3) a class of Käh1er manifolds not biho1omorphic to ℂn.KeywordsSectional CurvatureComplex ManifoldConformal TransformationRicci CurvatureConnection FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.