January 15 Abstract

We study the existence of special lagrangian submanifold using variational
approach. We prove that in a Kaehler-Einstein manifold, if a smooth
Lagrangian submanifold whose boundary lies in a complex hypersurface
minimizes volume among its Lagrangian relative homotopy class, then it has
mean curvature zero. And we prove a regularity theorem which concludes
that the area minimizer among a Lagrangian relative homotopy class is
smooth everywhere at the boundary.

Februry 5 Abstract

In the study of the gravitational radiation, Bondi associated to each null cone a number which is called the Bondi mass of the null cone. A natural question arises whether the physically reasonable gravitating systems can radiate more energy than they initially have,i.e., whether the Bondi mass can become negative.

Mathematically, it needs to establish a positive mass theorey for asymptotically hyperbolic 3-manifolds. The positivity of the Bondi mass was proved by Schoen and Yau in outline based on their positive mass argument around 1982. Almost at the same time, several physicists published the proofs of this positivity without providing mathematical detail based on Witten's spinor argument.

In this talk, we will discuss a definition of the energy-momenta for asymptotically hyperbolic 3-manifolds and the complete, rigious proof of the positivity of this energy-momenta. It is expected also this energy-momenta should have relation with the original definition of the Bondi mass."

Februry 26 Abstract

We will discuss some recent activities in classical and conformal differential geometry in connection to some fully nonlinear PDEs. These PDEs are in a form of the elementary symmetric functions of certain curvature tensors. In some aspect, they are similar to the Monge-Ampere equation.

One typical example is local version of the quermassintegrals of a convex body, which is called area measures. It can be expressed as a elementary symmetric function of principal radii of the boundary of the convex body. The classical Minkowski-Christoffel problem is concerning how to recover the convex body from one of its area measures. A similar problem was also posted for the elementary symmetric functions of principal curvatures by Alexandrov and Chern in 1950s. More recently, there are some new findings in connection to conformal geometry, as a fully nonlinear version of the Yamabe equation in the deformation of conformal metrics.

The algebra of the elementary symmetric functions, the analysis of the nonlinear PDEs and the geometry of curvature structures makes the subject interesting and challenging.