Harold Donnelly

Nodal Sets of Eigenfunctions: Riemannian Manifolds With Boundary

Let Hn+1 denote the simply connected complete space of constant curvature −1. The Laplacian Δ, acting on square integrable p-forms of H, is identified up to unitary equivalence.

Quantum unique ergodicity

Publisher Summary This chapter discusses the nodal sets of eigenfunctions. The basic tools for proving unique continuation theorems are Carleman inequalities. In his original paper, Aronszajn established such inequalities for second-order elliptic operators with C2,1 coefficients. Twice differentiable coefficients were needed for the use of geodesic polar coordinates. A substantial improvement was made in later joint work by Aronszajn, Krzywicki, and Szarski. They established strong unique continuation for second-order elliptic operators with C0,1 coefficients. In the Lipschitz case, one makes a preliminary conformal change in the metric associated to the second-order operator. This conformally changed metric has a coordinate representation with the necessary good properties of geodesic polar coordinates.

Essential spectrum and heat kernel

Consider a compact Riemannian manifold with ergodic geodesic flow. Quantum ergodicity is generalized from orthonormal bases of eigenfunctions of the Laplacian to packets of eigenfunctions. It is shown that this more general result is sharp. Namely, there may exist exceptional packets of eigenfunctions which concentrate on a submanifold.

Nodal domains and growth of harmonic functions on noncompact manifolds

Abstract Upper bounds are derived for the large time behavior of the heat kernel on Riemannian manifolds whose essential spectrum has a positive lower bound. Applications include a relative signature theorem and the construction of bounded harmonic functions.

COMPACT GROUP ACTIONS AND MAPS INTO ASPHERICAL MANIFOLDS

Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases.

Eigenforms of the laplacian on complete riemannian manifolds

A WELL-KNOWN theorem of Cartan and Hadamard states that a complete Riemannian n-manifold with everywhere nonpositive sectional curvature has a universal covering that is diffeomorphic to Euclidean n-space ([23, p. 184). This is one of many motivations for interest in spaces with contractible universal coverings; such spaces are called aspherical. Over the past dozen years considerable information has been obtained regarding group actions on such manifolds (compare[9-12]), and recently analogous questions for manifolds admitting suitably nontrivial maps into certain aspherical manifolds have been considered. In particular, the second named author has considered group actions on closed n-manifolds that map to the n-torus by a degree one map[22,23] and Schoen and Yau have considered smooth or real analytic actions on n-manifolds that- map to closed Riemannian manifolds of nonpositive curvature [21]. (Compare also [29]). In this paper, we shall prove results that overlap or contain the results of Borel-Conner-Raymond on aspherical manifolds, the results of [22], and the theorems of Schoen and Yau on group actions. One class,of results deals with manifolds that map onto a closed aspherical manifold by a map of nonzero degree; this class of manifolds contains all closed aspherical manifolds and in fact all connected sums of closed aspherical manifolds with other manifolds. The results of Borel, Conner and Raymond generalize completely; the only connected compact Lie groups that can act are tori, and all such actions are injective in the sense of [9]. In fact, if f : A4 + N is the map from M to the closed aspherical manifold N and i : T + M is an orbit map, then the composite is injective. Furthermore, if the map f has degree one, then (i) a finite group G acting on M with fixed points induces an injection from G to Aut(n,(M, WI,,)), where m. is a fixed point, (ii) if rr,(M, mo) is centerless, then there is an injection of G into the outer automorphism group of r,(M, mo) even if G has no fixed points. The results of [21] also indicate that one has strong restrictions on the differential symmetry of M, even if dim N > dim M, so long as f maps the fundamental class of M sufficiently nontrivially and N has nonpositive curvature. Our results show that similar restrictions also hold on the topological symmetry of M. Technically speaking, the central ideas of the proofs are (i) fairly explicit descriptions of the fundamental groups of certain orbit spaces (ii) some general facts about the fundamental groups of aspherical manifolds in general and nonpositively curved manifolds in particular. This paper consists of three sections. The first provides the necessary information on orbit spaces and their fundamental groups, while the second discusses group actions on manifolds with nonzero degree maps into aspherical manifolds and the third discusses group actions on manifolds with suitable maps into higher dimensional manifolds of nonpositive curvature.

Compactness of isospectral potentials

Soit M une variete de Riemann complete. Valeurs propres en dessous du spectre essentiel, estimations L 2 . Valeurs propres discretes. Estimation L infini des formes propres

Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

The Schrodinger operator -Δ+V, of a compact Riemannian manifold M, has pure point spectrum. Suppose that V 0 is a smooth reference potential. Various criteria are given which guarantee the compactness of all V satisfying spec(-Δ+V) = spec(-Δ+V 0 ). In particular, compactness is proved assuming an a priori bound on the W s,2 (M) norm of V, where s > n/2 - 2 and n = dim M. This improves earlier work of Bruning. An example involving singular potentials suggests that the condition s > n/2 - 2 is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions n ≤ 9.