stability, Energy Functional, and Kahler-Einstein Metrics

摘要

Starting with the work of Yau [Yl], Donaldson [Dl], and Uhlenbeck-Yau Y], the notion of stability has revealed itself under many guises to be closely related to the existence of canonical metrics in Kahler geometry. The equivalence between Hermitian-Einstein metrics on vector bundles and Mumford stability was proved by Donaldson and Uhlenbeck-Yau in [D1] and [UY], while the existence of Kahler-Einstein metrics was conjectured in the early 1980's by Yau [Y2] to be equivalent to stability in geometric invariant theory. At the present time, the Yau conjecture has been at least partially confirmed. The existence of Kahler-Einstein metrics has been shown to imply if-stability and CM-stability by Tian [T2], and more recently to imply how-Mumford stability by Donaldson [D2j.