Energy properness and Sasakian-Einstein metrics (Q635843)

The result obtained by \textit{G. Tian} in [Invent. Math. 130, No.~1, 1--37 (1997; Zbl 0892.53027)] (concerning the existence of the Kähler-Einstein metric) is generalized here. On a compact Sasaki manifold, some necessary and sufficient conditions (closely related to the properness of the corresponding energy functional) are given for the existence of a Sasaki-Einstein metric. Under the condition of admitting no nontrivial Hamiltonian holomorphic vector fields, the author proves that the existence of a Sasaki-Einstein metric implies a Moser-Trudinger type inequality. A Miyaoka-Yau type inequality is obtained at the end.