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 and construct 2 types of metric cones, which are natural analogues of the Kähler case. We introduce a new type of fully nonlinear geometric PDE on such surfaces and establish smooth solvability. As a geometric application, we solve the prescribed Bismut–Ricci problem. We also introduce a weighted version of this problem. With this, we obtain canonical metrics on two important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">S</m:mi> <m:mi>M</m:mi> </m:msub> </m:math> \mathcal{S}_{M} .