## Tohoku Mathematical Journal 2002 June SECOND SERIES VOL. 54, NO. 2

 Tohoku Math. J. 54 (2002), 259-275

Title HEAT KERNEL ESTIMATES AND THE GREEN FUNCTIONS ON MULTIPLIER HERMITIAN MANIFOLDS

Author Toshiki Mabuchi

(Received June 7, 2000)
Abstract. Using a standard technique of Li and Yau, we study heat kernel estimates for a special type of compact conformally Kahler manifold, called a multiplier Hermitian manifold of type $\sigma$, which we derive from a Hamiltonian holomorphic vector field on the manifold. In particular, we obtain a lower bound estimate for the Green function averaged by the associated group action. For a fixed $\sigma$, such an estimate is known to play a crucial role in the proof of the uniqueness, modulo a group action, of Einstein multiplier Hermitian structures on a given Fano manifold.

2000 Mathematics Subject Classification. Primary 32W30; secondary 53C55, 14J45, 14J50.