Speaker: 许斌（中国科学技术大学）

Time: 10:00-11:00, Jun. 14

Venue: 数学中心研讨室（至善楼602）

Title: THE SPACE OF GERMS OF EXTREMAL KA ̈HLER METRICS IN ONE DIMENSION COMPRISES THREE DISTINCT R^3 COMPONENTS

Abstract:

In the 1980s, Eugenio Calabi introduced the concept of ex- tremal K ̈ahler metrics as critical points of the L2-norm functional of scalar curvature in the space of K ̈ahler metrics belonging to a fixed K ̈ahler class of a compact complex manifold X. Calabi demonstrated that extremal Ka ̈hler metrics always degenerate into Einstein metrics on compact Riemann surfaces. We define a K ̈ahler metric g on a domain of Cn as a local extremal K ̈ahler metric of dimension n if it satisfies the Euler-Lagrange equation of this functional, i.e. holomorphic is the (1, 0)-part of the gradient vector field of the scalar curvature of g, in the domain. Our main result establishes that the space of all germs of local extremal, non-Einstein Ka ̈hler metrics of dimension one comprises three components, each diffeomorphic to R^3.