ar X iv : m at h / 02 03 25 4 v 1 [ m at h . D G ] 2 4 M ar 2 00 2 STABILITY , ENERGY FUNCTIONALS , AND KÄHLER - EINSTEIN METRICS

Starting with the work of Yau [Y1], Donaldson [D1], and Uhlenbeck-Yau [UY], the notion of stability has revealed itself under many guises to be closely related to the existence of canonical metrics in Kähler 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 Kähler-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 Kähler-Einstein metrics has been shown to imply K-stability and CM -stability by Tian [T2], and more recently to imply Chow-Mumford stability by Donaldson [D2].