Conformally invariant powers of the Laplacian -- A complete nonexistence theorem

We show that on conformal manifolds of even dimension there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian for n/2$">. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of for , is sharp.

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds

Assume is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case . We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.

     

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken -manifolds

Understanding non-Haken -manifolds is central to many current endeavors in -manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the -connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.

     

Nonzero degree maps between closed orientable three-manifolds

This paper adresses the following problem: Given a closed orientable three-manifold , are there at most finitely many closed orientable three-manifolds 1-dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

     

The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature

We study the class of closed -dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally -connected. We also study the limit of -manifolds with -curvature bound for .

     

Hamilton's gradient estimate for the heat kernel on complete manifolds

In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with . We accomplish this extension via a maximum principle of L. Karp and P. Li and a Berstein-type estimate on the gradient of the solution. An application of our result, together with the bounds of P. Li and S.T. Yau, yields an estimate on the gradient of the heat kernel for complete manifolds with non-negative Ricci curvature that is sharp in the order of for the heat kernel on .

     

Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of on , and we show that the quotients of fibers by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on .

We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

     

Differentiable mappings with an infinite number of critical points

In this paper we shall give some sufficient conditions in order that the so-called -category of a pair of differentiable manifolds be infinite.

     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

     

Stably and almost complex structures on bounded flag manifolds

We study the enumeration problem of stably complex structures on bounded flag manifolds arising from omniorientations, and determine those induced by almost complex structures. We also enumerate the stably complex structures on these manifolds which bound, therefore representing zero in the complex cobordism ring .

     

Four-Manifolds With Surface Fundamental Groups

We study the homotopy type of closed connected topological -manifolds whose fundamental group is that of an aspherical surface . Then we use surgery theory to show that these manifolds are -cobordant to connected sums of simply-connected manifolds with an -bundle over .

     

Uniqueness of the Kontsevich-Vishik trace

Let be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on , whose (complex) order is not an integer greater than or equal to , is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the -operator trace on trace class operators.

Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique.

     

The -harmonic forms on rotationally symmetric Riemannian manifolds revisited

We use separation of variables for generalized Dirac operators on rotationally symmetric Riemannian manifolds to recover a theorem of Dodziuk regarding the spaces of -harmonic forms on such manifolds.

     

Supercongruences between truncated hypergeometric functions and their Gaussian analogs

Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.

     

The remainder in Weyl's law for -dimensional Heisenberg manifolds

We prove that the error term in Weyl's law for `rational' -dimensional Heisenberg manifolds is of order . In the `irrational' case, for generic -dimensional Heisenberg manifolds with 1$">, we prove that the error term is of the order . The polynomial growth is optimal.

     

Examples of non-formal closed -connected manifolds of dimensions

We construct closed -connected manifolds of dimensions that possess non-trivial rational Massey triple products. We also construct examples of manifolds such that all the cup-products of elements of vanish, while the group is generated by Massey products: such examples are useful for the theory of systols.

     

Branched coverings and nonzero degree maps between Seifert manifolds

In this note we give a necessary and sufficient condition for the existence of a fiber preserving branched covering between two closed, orientable Seifert manifolds (for sufficiency we need the additional assumption that the genus of the base orbifold of the target manifold ). Combining this with two theorems of Rong we get a necessary and sufficient condition for the existence of a nonzero degree map between two such manifolds.

     

On complete holomorphic vector fields

We show that, on holomorphic manifolds that have a plurisubharmonic exhaustion function and that do not carry nonconstant bounded plurisubharmonic functions (e.g. ), holomorphic vector fields that are complete in positive time are complete in complex time.

     

Legendrian contact homology in

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form , where is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of and, more generally, invariants of self transverse immersions into up to restricted regular homotopies. When , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.