Fully nonlinear flows and Hessian equations on compact Kahler manifolds

In this thesis, we will study a class of fully nonlinear flows on Kähler manifolds. This family of flows generalizes the previously studied J-flow. We use the quotients of elementary symmetric polynomials or log of them to construct the flow. We obtain a necessary and sufficient condition in terms of positivity of certain cohomology class to guarantee the convergence of the flow. The corresponding limit metric gives rise to a critical metric satisfying a Hessian type equation on the manifold. We shall also discuss several geometric applications of our main result.