TITLES and ABSTRACTS

Haizhong Li (Tsinghua University)
Title: Embedded constant mean curvature tori in the three-sphere
Abstract: The study of constant mean curvature (CMC) surfaces in spaces of constant curvature (that is, R^3, sphere S^3, and hyperbolic space H^3), is one of the classical subjects in differential geometry. There are many beautiful results on this topic. By constructing a holomorphic quadratic differential for CMC surfaces, H. Hopf showed that any CMC two-sphere in R^3 is a round sphere. Then S. S. Chern extended Hopf's result to CMC two-spheres in 3-dimensional space forms. The famous Hopf conjecture asks that whether a compact CMC surface in R^3 is necessarily a round sphere? In 1980s, H. C. Wente gave a counter example of this conjecture by constructing a compact immersed CMC torus in R^3. In 1950s, A. D. Alexandrov showed that if a compact CMC surface is embedded in R^3, H^3 or a hemisphere S^3_+, then it must be totally umbilical. The minimal surface is the surface with constant mean curvature zero. It was conjectured by H. B. Lawson in 1970s that the only embedded minimal torus in three-sphere is the Clifford torus. In 1980s, U. Pinkall and I. Sterling conjectured that embedded tori with CMC in three-sphere are surfaces of revolution. Recently Simon Brendle solved the Lawson conjecture. Then Ben Andrews and I gave a complete classifcation of CMC embedded tori in the three-sphere. When the constant mean curvature is non-positive or 1/sqrt{3}, the only embedded torus is the Clifford torus. For other values of the mean curvature, there exists embedded torus which is not the Clifford torus, and Ben Andrews and I give a complete description of such surfaces. As a Corollary, Andrews-Li's Theorem solves the famous Pinkall-Sterling conjecture. In this lecture, we will talk about some basic facts of CMC surfaces and explanations of key ideas of proofs of Lawson conjecture and Pinkall-Sterling conjecture.

Reiko Miyaoka (Tohoku University)
Title: Hypersurface geometry, classical and modern
Abstract: My talk plan is as follows (possibly with a slight change):
1. Generalizing the surface theory, we give fundamental facts on hypersurface geometry, mainly in the space forms. Hypersurfaces given by tubes over lower dimensional submanifolds are especially our concern.
2. We introduce special hypersurfaces such as Dupin and isoparametric hypersurfaces. Topology in the compact case is important.
3. We introduce Lie sphere geometry extended from Moebius geometry. Then we investigate Dupin hypersurfaces in relation with isoparametric hypersurfaces.
4. We give the latest result on the classification of isoparametric hypersurfaces in S^n, and give an expression of the defining polynomials in terms of the moment map of certain group actions.

Pascal Romon (Universite Paris-Est)
Title: Metric, complex and symplectic structures on the tangent bundle
Abstract: When M is a differential manifold, its tangent bundle TM is a also differential manifold, of twice the dimension of M. What kind of structure can TM be endowed with? For example, if M is riemannian, is there a "natural" metric on TM?
We will inspect three different but related type of structures: metric (riemannian or pseudo-riemannian), symplectic and complex or almost-complex (paracomplex if times allows), and see whether such structures exist on TM, possibily depending on analogous structures on M (and sometimes not). We will start by recalling the definitions of these structures, and then move on definitions on the tangent bundle. We will end up by stating recent results and applications of this problem to geometric questions.

Felix Schulze (Frei Universitaet Berlin)
Title: Singularity formation and resolution in mean curvature flow
Abstract: In this series of lectures we will introduce mean curvature flow of hypersurfaces in Euclidean space and of networks in the plane. We will discuss the fundamental results of Huisken that convex surfaces contract to 'round' points in finite timeand of Gage/Hamilton and Grayson that closed, embedded curves contract to 'round' points in finite time as well.
To do this we will investigate the implications of Huisken's monotonicity formula and study self-similarly contracting solutions. These special solutions can also be seen as tangent flows to mean curvature flow. Tangent flows arise as subsequential parabolic rescalings of the flow at a fixed point in space-time. In the case that such a tangent flow consists of a compact, smooth, unit density self-similarly shrinking solution we will also show that at such a point the tangent flow is unique, i.e. it does not depend on the chosen sequence of rescalings. Furthermore, we will the study the resolution of initial conical singularities in the flow of networks in the plane.