UC Irvine

We consider various geometric flows which are well adapted for the study of non-K\"ahler complex manifolds. We first study solutions to the pluriclosed flow on compact complex surfaces, giving a complete classification of the long time behavior of homogeneous solutions to the flow and constructing non-trivial, non-K\"ahler expanding soliton solutions to the pluriclosed flow. We also give a simple expression for the evolution of the Lee form on a complex surface, and use this to give simplified proofs of various classification results for fixed points of the flow.

We also consider a functional which is closely related to the Dirichlet energy of maps between two Hermitian manifolds and which has holomorphic maps as global minimizers. We derive it's first and second variations and consider the associated parabolic flow. We provide conditions under which the flow converges to a critical point of the functional and give explicit examples of nice solutions to the flow in Hopf surfaces. We additionally demonstrate so called 'bubbling' criteria for solutions of the flow on surfaces and, using this functional, we give a variational proof that submanifolds of Vaisman manifolds are Vaisman.