Differential Geometry

We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S N ∖Λ , where Λ is a disjoint union of submanifolds of dimensions between 0 and (N−2)/2 . The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two 4 -manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the 4 -manifold resulting as a connected sum along the surface. We study examples in which this is enough to describe completely the basic classes.

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields on three-manifolds. Thus, we characterise Beltrami fields in a metric-independant manner. This correspondence yields a hydrodynamical reformulation of the Weinstein Conjecture, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all C ∞ rotational Beltrami flows on S 3 . This is the key step for a positive solution to the hydrodynamical Seifert Conjecture: all C ω steady state flows of a perfect incompressible fluid on S 3 possess closed flowlines. In the case of Euler flows on T 3 , we give general conditions for closed flowlines derived from the homotopy data of the normal bundle to the flow.

The purpose of this paper is to present the first continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, and simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. The examples presented here are Riemannian three-step nilmanifolds and thus provide a counterexample to the Ouyang-Pesce Conjecture for higher-step nilmanifolds. Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method for constructing isospectral nilmanifolds. In particular, all continuous families of Riemannian two-step nilmanifolds that are isospectral on functions must also be isospectral on p-forms for all p. They conjectured that all isospectral deformations of nilmanifolds must arise in this manner. These examples arise from a general method for constructing isospectral Riemannian nilmanifolds previously introduced by the author.

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (Neumann and Dirichlet) isospectral metrics which have different local geometry on manifolds with boundary in every dimension greater than 6 and also new examples of pairs of closed isospectral manifolds with different local geometry. These examples illustrate for the first time that the Ricci curvature of a Riemannian manifold is not spectrally determined.

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.

We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.

We relate Kostant's theorem on the cohomology of a flag manifold G/B with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisson structure. We do this by writing everything down explicitly in some coordinates on each cell.

Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L 2 torsion, which lies in the determinant line of the twisted L 2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in an earlier paper by the authors. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic L 2 torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic L 2 torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

This contains a list of (mostly very minor) corrections to the book Introduction to Symplectic Topology, Clarendon Press, Oxford, (1995), together with rewritten versions of two lemmas and some additional comments.