Weiyong He

On the Calabi Flow

In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c3,α(M) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kähler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kähler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.

On the transverse Scalar Curvature of a Compact Sasaki Manifold

Abstract We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group

Second-Order Geometric Flows on Foliated Manifolds

We prove a general result about the short-time existence and uniqueness of second-order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in the literature, as the transverse Ricci flow, the Sasaki–Ricci flow and the SasakiJ-flow and motivate the study of other evolution equations. We also introduce a transverse version of the Kähler–Ricci flow adapting some classical results to the foliated case.

On the regularity of the complex Monge-Ampère equations

We shall consider the regularity of solutions for the complex Monge-Ampère equations in Cn or a bounded domain. First we prove interior C2 estimates of solutions in a bounded domain for the complex Monge-Ampère equations with the assumption of an Lp bound for u, p > n2, and of a Lipschitz condition on the right-hand side. Then we shall construct a family of Pogorelov-type examples for the complex Monge-Ampère equations. These examples give generalized entire solutions (as well as viscosity solutions) of the complex Monge-Ampère equation det(uij̄) = 1 in C n.