Generalized Kahler-Einstein Metrics and Energy Functionals

In this paper, we consider a generalized Kahler-Einstein equation on a Kahler manifold M. Using the twisted X-energy introduced by Song and Tian, we show that the existence of generalized Kahler-Einstein metrics with semi-positive twisting (1, 1)-form theta is also closely related to the properness of the twisted X-energy functional. Under the condition that the twisting form theta is strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kahler Einstein metric implies a Moser-Trudinger type inequality.