Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on closed manifold $M$, where $g$ is Riemannian metric and $X$ 1-form. It known that when the cosmological constant vanishes, there rigidity: vanishes consequently Ricci flat. We study this form rigidity for all signs constant. has been asserted also holds negative, but we exhibit counter-example....

– in this paper we consider some (α ,β ) -metrics such as generalized kropina, matsumoto and f (α β )2α = + metrics, and obtain necessary and sufficient conditions for them to be einstein metrics when βis a constant killing form. then we prove with this assumption that the mentioned einstein metrics must beriemannian or ricci flat.

This article is a summary of some of the author’s work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vice versa. Despite this physical motivation, many recent results are of indepe...

A log del Pezzo surface is a projective surface with quotient singularities such that its anticanonical class is ample. Such surfaces arise naturally in many different contexts, for instance in connection with affine surfaces [Miyanishi81], moduli of surfaces of general type [Alexeev94], 3 and 4 dimensional minimal model program [Alexeev93]. They also provide a natural testing ground for existe...