报告题目：Positive mass theorem in several complex variables
报告： 郑日新 教授(中研院数学所)
报告摘要： Inspired by the ideas in general relativity, we define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (of a certain class, modelling on the boundary of a bounded strongly pseudoconvex domain) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. For compact spherical CR manifolds of dimension $\geq 7$ (dimension 5 with an extra technical condition, resp.) with positive CR Yamabe invariant, we obtain a positive mass theorem (PMT in short) through a method of comparing Green’s functions. We can also study the PMT through a spinor approach. The Lichnerowicz - type formula contains a trace curvature term and a $T$-derivative term besides typical terms as in the Riemannian situation. In dimension 5 it happens that no $T$-derivative term is involved. By assuming the existence of spin structure, we can get rid of the trace curvature term. Thus we obtain a PMT for 5-dimensional compact spin, spherical CR manifolds with positive Yamabe invariant. In dimension 3, the trace curvature term is absorbed in the scalar curvature term while the nonnegativity of the Paneitz term associated to the $T$-derivative becomes a condition to have a PMT in dimension 3. We apply the PMT to find solutions of the $CR$ Yamabe problem with minimal energy and to prove the convergence of the CR Yamabe flow. The PMT for dimension $\geq 7$ (5, 3, resp.) is joint work with Hung-Lin Chiu and Paul Yang (Hung-Lin Chiu, Andrea Malchiodi and Paul Yang, resp.).
报告人简介： Dr. Cheng's main research interest is on Cauchy-Riemann(CR for short) geometry and contact topology. In his Ph.D. dissertation, he classified all possible Lie algebras with the 2-gradation property-which is a special feature of CR-structure. Later he proved an equivalence between CR geometry and geometry of chains, the distinguished invariant curves in CR geometry. In 1986 he got a NSC (National Science Council) grant for a one-year at the Harvard University. At Harvard, Professor Siu introduced John Lee to him. There John Lee and he started the study of various kinds of moduli spaces of CR structrues in dimension 3. In the fall of 1992, he had a second visit supported by the NSC. This time he visited Professor Eliashberg of the Stanford University and began to realize the deep connection between contact topology and CR geometry and their roles in the study of 3-manifolds. Many interesting problems in this respect are just emerging and are expected to attract his attention for the following years.