Convergence of a class of fully non-linear parabolic equations on Hermitian manifolds

TL;DR

This paper studies a class of fully non-linear parabolic equations on compact Hermitian manifolds, proving long-term existence and convergence of solutions, and establishing a Harnack inequality for the linearized equations.

Contribution

It introduces new convergence results for non-linear parabolic equations on Hermitian manifolds and derives a Harnack inequality for the linearized form.

Findings

Solutions exist for all time and converge under general conditions.

A Harnack inequality for the linearized equation is established.

The results extend understanding of non-linear parabolic equations on complex manifolds.

Abstract

We consider a class of fully non-linear parabolic equations on compact Hermitian manifolds involving symmetric functions of partial Laplacians. Under fairly general assumptions, we show the long time existence and convergence of solutions. We also derive a Harnack inequality for the linearized equation which is used in the proof of convergence.