On canonical metrics of complex surfaces with split tangent and related geometric PDEs

Abstract

In this paper, we study bi-Hermitian metrics on complex surfaces with split holomorphic tangent bundle and construct 2 types of metric cones. We introduce a new type of fully non-linear geometric PDE on such surfaces and establish smooth solutions. As a geometric application, we solve the prescribed Bismut Ricci problem. In various settings, we obtain canonical metrics on 2 important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type ${\mathcal{S}}_M$.

41 pages

References in corpus (2)

• Generalized Calabi-Yau metric and Generalized Monge-Ampere equation
• Leafwise flat forms on Inoue-Bombieri surfaces

Cited by in corpus (1)

• The parabolic split-type Monge-Ampere on split tangent bundle surfaces