Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are 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. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method. To the memory of Hubert Pesce, a valued friend and colleague.     

Manifolds with quadratic curvature decay and fast volume growth

 We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms. Received: 21 March 2001 / Revised version: 21 March 2002 / Published online: 10 February 2003 Research partially supported by NSF grant DMS-0072154 and MSRI     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

Length spectra and<Emphasis Type="Italic">p</Emphasis>-spectra of compact flat manifolds

We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.     

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. In memory of Robert Brooks     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Spectral Bounds on Orbifold Isotropy

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.This research was partially supported by NSF grant DMS 0072534.     

Higher degree layer potentials for non-smooth domains with arbitrary topology

We consider layer potentials associated with the Hodge-Laplacian acting on differential forms of arbitrary degree defined on Lipschitz subdomains of a Riemannian manifold. The main emphasis is on the interplay between the mapping properties of such layer potentials and the topology of the underlying domain.Partially supported by a UMC Research Board GrantPartially supported by NSF grant DMS-9870018     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

Polynomial averages converge to the product of integrals

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL ² to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

Initial segments of the degrees of constructibility

We show that any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension ofL. Research partially supported by NSF grant DMS-8601777. Research partially supported by NSF grant DMS-8601048 and grant 84-00067 from the U.S.-Israel Binational Science Foundation. Thanks are also due to Uri Abraham and Mati Rubin for helpful discussions about initial segments of the degrees of constructibility and to Bill Lampe for information on algebraic lattices.     

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

We provide a very general result which identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover within a unified framework the HVZ Theorem and Krein's results on orthogonal polynomials with finite essential spectra. Supported in part by The Israel Science Foundation (grant No. 188/02). Supported in part by NSF grant DMS-01 40592. Research supported in part by grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

The Lévy-Hinčin representation for random compact convex subsets which are infinitely divisible under Minkowski addition

Summary This paper establishes the Lévy-Hinčin representation for all random compact convex subsets of ℝ which are infinitely divisible for Minkowski addition. Research partially supported by NSF grants No. MCS 8100728 and DMS-8318610 Research partially supported by NSF grant No. MCS 8301326     

Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.Partially supported by NSF grant MCS77-18723(02).