We study the renormalized volume of a conformally compact Einstein manifold. In even dimenions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the Q-curvature. 0. Introduction Recently, there is a series of work ([GZ],[FG-2] and [FH]) exploring the connecti...

We find the necessary and sufficient conditions for a sequential warped product manifold to be quasi-Einstein manifold. also investigate standard static space-time generalized Robertson-Walker of quasi-constant curvature.

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality. A smooth Riemannian manifold (M, g) is said [1] to be Einstein if its Ricci curvature is a constant multiple of g. Any i...

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric onX\D with cusp singularity alongD. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application ...

We characterize the Einstein metrics in such broad classes of as almost $$\eta $$ -Ricci solitons and on Kenmotsu manifolds, generalize some known results. First, we prove that a metric an soliton is if either it -Einstein or potential vector field V infinitesimal contact transformation collinear to Reeb field. Further, manifold admits gradient with leaving scalar curvature invariant, then mani...

The main result of this paper is that the space of conformally compact Einstein metrics on any given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric a...

LetM be an (n+1)-dimensional manifold with non-empty boundary, satisfying π1(M,∂M) = 0. The main result of this paper is that the space of conformally compact Einstein metrics on M is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with p...

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kähler-Einstein manifolds. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in ...

We show that #9(S 2 S 3) admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the rst known Einstein metrics on this 5-manifold. In particular, the bound b 2 (M)8 which holds for any regular Sasakian-Einstein M does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and ...