We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

First, we show that a warped product of line with Riemannian manifold (fiber) is weakly conformally flat and quasi Einstein if only the fiber in which case Bach flat. Finally, characterize classify contact manifolds satisfying (and doubly weakly) quasi-Einstein ( $$\eta $$ -Einstein) conditions.

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connectio...

This paper is involved with the asymptotics associated to Bohr-Sommerfeld Lagrangian submanifolds of a compact Hodge manifold, in the context of geometric quantization (see e.g. [BW], [BPU], [GS3], [W]). We adopt the general framework for quantizing Bohr-Sommerfeld Lagrangian submanifolds presented in [BPU], based on applying the Szegö kernel of the quantizing line bundle to certain delta funct...

We prove that if (φn) ∞ n=0, φ0 ≡ 1, is a basis in the space of entire functions of d complex variables, d ≥ 1, then for every compact K ⊂ C there is a compact K1 ⊃ K such that for every entire function f = ∑∞ n=0 fnφn we have ∑∞ n=0 |fn| supK |φn| ≤ supK1 |f |. A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.

It is still very poorly understood which 5-manifolds carry an Einstein metric with positive constant. By Myers' theorem, the fundamental group of such a manifold is finite, therefore it is reasonable to concentrate on the simply connected case. The most familiar examples are connected sums of k copies of S 2 × S 3. For k ≤ 9, Einstein metrics on these were constructed by Boyer, Galicki and Naka...

Let M be a compact complex manifold of complex dimension two with a smooth Kahler metric and D a smooth divisor on M. If E is a rank 2 holomorphic vector bundle on M with a stable parabolic structure along D, we prove that there exists a Hermitian-Einstein metric on E' = E|M\D compatible with the parabolic structure, and whose curvature is square integrable. MIRAMARE TRIESTE January 2000

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connectio...

According to the conjecture of Calabi, on a complex manifold X with ample canonical bundle KX , there should exist a Kähler-Einstein metric g. Namely, a metric satisfying Ricg = −ωg, where ωg is the Kähler form of the Kähler metric g. The existence of such metric when X is compact was proved by Aubin and Yau ([23]) using complex Monge-Ampère equation. This important result has many applications...

A para-Kähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. a parallel field of skew-symmetric endomorphisms with K = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie gro...