Full proofs of results announced here are to be found in [lO]. We consider the problem of classifying germs of neighbourhoods of locally flat topological submanifolds. Let M be a fixed topological manifold. An r-neighbourhood of M is a pair (i, N) where N is a topological manifold of dimension n+r and i:M—*N a locally flat embedding satisfying i^ÇdN) —dM. Denote the set of germs of r-neighbourh...

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set of small surface area. These estimates involve either a volume bound or a spec...

The classical boundary-value problem for positive-definite solutions of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact (S) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii (a, b). For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found ...

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry ...

We consider the T–equivariant cohomology of Bott–Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi–simple complex algebraic group of adjoint type with maximal torus T . We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of the associated Weyl group such that its global sections give the T–equivaria...

We show that solutions of the Seiberg-Witten equations lead to nontrivial estimates for the L2-norm of the Weyl curvature of a compact Riemannian 4-manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics on smooth compact 4-manifolds with a non-zero Seiberg-Witten invariant. These results considerably refine those previously obtained [21] by using...

A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic manifold X with zero first Chern class and a Kähler form ω ∈ H(X,Z). The form ω is called a polarization. Let M be the universal deformation space of (X,ω). M is smooth by a theorem of Tian [5]. By [8], we may assume that each X ′ ∈ M is a Kähler-Einstein manifold. i.e. the associated Kähler metric (g′ αβ ) is Ricci flat. The

Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci ...

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds X of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (“Weyl-type”) saving of vol(X)...