UC Irvine

We study some selected topics on Hessian type equations.

In the first chapter, our goal to generalize the quantitative version of the constant rank theorem by Sz {e}kelyhidi-Weinkove onto Hermitian manifolds and the complex coordinate space. As an application, we also study some properties of Ricci tensors based on the theorem we get.

In the second chapter, we consider C² estimates for complex Hessian equations involving gradient terms. In particular, we study special cases when the eigenvalues are bounded below.

In the third chapter, we study the long-time existence and convergence of parabolic complex Hessian type equations whose second order operator is not necessarily convex or concave.