One of the natural generalizations of conformal structure on a two dimensional surface is a conformally flat structure on an n-manifold. In higher dimensions, not every manifold admits such a structure and it is a difficult problem to give a good classification of conformally flat manifolds. Recall that conformally flat Riemannian manifolds are manifolds whose metrics are locally conformally equivalent to the Euclidean metric. Kuiper [Kul, Ku2] was the first to study the global properties of these manifolds. He classified those compact conformally flat manifolds with abelian fundamental group. For dimensions greater than two, the Liouville theorem tells us that the conformal transformations of S" are determined locally and are given by M6bius transformations. Hence by a standard monodromy argument, a simply connected conformally flat manifold (with dimension> 3) has a conformal immersion into S" which is unique up to

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.

In this paper, we study the behavior of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.

In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four- manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.

If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that 1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is M¨obius-like but not conjugate into PSL(2,R).We conjecture that the latter possibility cannot occur.