Thomas Kwok-keung Au

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

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.

Off-Center Reflections: Caustics and Chaos

We study the possible link between the dynamics of a certain family of circle maps and the caustics of their iterates. The maps are defined by off-center reflections in a mirrored circle; they can also be regarded as perturbed rotations. Someof our experimental observations can be justified rigorously: for example, a lower bound is given for the number of cusps and the modelock ing behavior are studied. Symplectic topology is a particularly useful tool in this study.

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.