Generalised Einstein equations and prescribed relations for the first Chern Weil form
The author considers generalizations of the Einstein equation on a (closed) Kähler manifold \(V\) with a fixed Kähler form \(\omega\). To describe it, let \(\lambda\) be \(-1,0,1\) according to the curvature of the anticanonical bundle of \(V\) and \(\omega_x=\omega +\sqrt{-1}\partial \overline\partial x\) for a function \(x\) belonging to a Banach space of functions on \(V\) in which the test functions are dense. The equation discussed in the paper is \(\sqrt{-1}\text{Ric}(\omega_x)-\lambda\omega_x= \sqrt{-1}\partial\overline \partial((t-\lambda)x-sz)\), where \(z\) is a given function on \(V\) and \(t,s\) are real parameters. The main results of the paper (Theorems 1.1, 1.2) give sufficient conditions for the existence and unicity of solutions of certain regularity.
scientific article; zbMATH DE number 2093681
scientific article