Tom Yau-heng Wan

Parabolic constant mean curvature spacelike surfaces

In this paper, we classify Lorentzian isometry classes of parabolic constant mean curvature cuts by conformal classes of nonzero holomorphic quadratic differentials on C

Vanishing Theorems for Conformally Compact Manifolds

Abstract We study L 2 harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to −1 and the infimum of the L 2 spectrum of the Laplacian great than p(n − p) for some p ≤ n/2, then there is no nontrivial L 2 harmonic p-form.

Analysis on an ODE arisen from studying the shape of a red blood cell

We study the existence problem of a special solution to the Helfrich functional such that it corresponds to a surface of the shape of a red blood cell. The Helfrich functional is also a perturbation of the Willmore functional involving some parameters with physical meanings. With the expected symmetry of the surface, it reduces to an analysis on an ODE with certain shape requirements. We discover a sufficient condition on the parameters which ensures the existence of such a special solution to the ODE.

Hyper-Lagrangian Submanifolds of Hyperkähler Manifolds and Mean Curvature Flow

In this article, we define a new class of middle dimensional submanifolds of a Hyperkähler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation. It is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.