On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds

Abstract

The thesis consists of two parts. In the first part of the thesis, we classify the convergence behaviour of rotationally symmetric solutions to the modified J-flow on the blow-up of the complex projective space at a point. In the second part of the thesis, we adapt the PDE approach of Guo et al. to prove an L∞ estimate for transverse complex Monge-Amp`ere equations on a homologically orientable transverse Kahler manifolds. As an application, we obtain a purely PDE proof of the regularity of Calabi-Yau cone metrics on Q-Gorenstein T-varieties.