Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?     

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C^∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S¹, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.     

The wave trace of the basic Laplacian of a Riemannian foliation near a non-zero period

Given a compact boundaryless Riemannian manifold that admits a Riemannian foliation, recall that the space of leaf closures is a singular stratified space. Associated to this space is an operator called the basic Laplacian defined on the space of smooth functions that are constant on the leaves (and, hence, the closures of the leaves of the foliation). The corresponding basic spectrum is, under certain assumptions, an infinite subset of the spectrum of the ordinary laplacian. If the metric is bundle-like with respect to the foliation, the trace of the basic wave operator can be analyzed, and invariants of the basic spectrum can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the leaf closures, and from them, basic wave trace asymptotic expansions are derived. Using the connection between Riemannian foliations and manifolds being acted upon by a compact Lie group of isometries, $G$ , the wave trace for the $G$ -invariant spectrum of a $G$ -manifold is sketched out as a related result.     

Embedding solenoids in foliations

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H¹(L;R) not equal to 0 and if the foliation is a product foliation in some saturated open neighborhood U of L, then there exists a foliation F^′ on M which is C¹-close to F, and F^′ has an uncountable set of solenoidal minimal sets contained in U that are pairwise non-homeomorphic. If H¹(L;R) is 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighborhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.     

The propagation of singularities along gliding rays

Summary LetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator such that the boundary value problem (P, B) is normal and satisfies the Lopatinskii-Schapiro condition. The singularities of distributions,u, such thatP u is smooth on the boundary, near points at which the boundary is bicharacteristically convex are shown to propagate, in the boundary, only along the gliding rays, which are the leaves of the Hamilton foliation of the glancing surface. This analysis, combined with known results on diffraction, leads to a Poisson relation bounding the singular support of the Fourier transform of the Dirichlet spectral density for a compact Riemannian manifold with geodesically convex, or concave, boundary in terms of the geodesic length spectrum.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

The Dirac spectrum on manifolds with gradient conformal vector fields

We show that the Dirac operator on a spin manifold does not admit L² eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.     

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spin ^c Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

Coercive singular perturbations II: Reduction and convergence

An algebra of pseudodifferential singular perturbations is introduced. It provides a constructive machinery in order to reduce an elliptic singularly perturbed operator (in $\text{R}$ⁿ or on a smooth manifold without border) to a regular perturbation. The technique developed is applied to some singularly perturbed boundary value problems as well. Special attention is given to a singular perturbation appearing in the linear theory of thin elastic plates. A Wiener-Hopf-type operator containing the small parameter reduces this singular perturbation to a regular one. It also gives rise to a natural recurrence process for the construction of high-order asymptotic formulae for the solution of the perturbed problem. The method presented can be extended to the general coercive singular perturbations.     

Spectrum and continuum eigenfunctions of Schrödinger operators

We consider Schrödinger operators $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ for a large class of potentials. V. We show that if H? = E? has a polynomially bounded solution ? then E is in the spectrum of H. This is accomplished by proving that the spectrum of H as an operator on L² is identical to its spectrum as an operator on the weighted L² space, L²_δ.     

A plane foliation of degree different from 1 is determined by its singular scheme

We prove that a foliation _877-1 of degree ≠ 1 on P² is completely determined by its singular subscheme SingS(_877-2) of P². We apply this result to obtain a similar characterization of _877-3 in terms of the configuration of base points associated to its singular scheme, in case every singularity of _877-4 has non-trivial linear part. Our main motivation comes from a well-known fact: in case a foliation _877-5 of degree r ≥ 2 on Pn has only isolated singularities of multiplicity 1, then _877-6 is completely determined by its singular set Sing(_877-7).     

THE LEGENDRE TYPE OPERATOR IN A LEFT DEFINITE HILBERT SPACE

Abstract

The techniques for discussing linear differential operators in left definite spaces, developed earlier for regular fourth order and singular second order operators, are applied the Legendre type operator. It is shown that the operator, with its domain merely restricted to the new space, remains self-adjoint and has the same spectrum, inverse and spectral resolution (an eigenfunction expansion) as the original L ² operator.     

On the basis property of eigenfunctions of a singular second-order differential operator

We consider the first boundary value problem for a singular differential operator of second order on an interval with transmission conditions at an interior point of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, 1) and forms a Riesz basis in that space.     

Note on a theorem of H. Moscovici

Let Γ be a discrete subgroup of a semisimple Lie group G such that ΓβG has a finite volume. Using a theorem of Moscovici we express the multiplicity of discrete series representations of G in the discrete spectrum of L²(ΓβG) as the L²-index of a twisted Dirac operator. This result, which extends a result of Moscovici and of the author, holds for all integrable discrete series and for infinitely many nonintegrable discrete series. In particular, up to computing L²-indices in the special rank one case, it implies the Osborne-Warner formula.     

Vanishing theorem for transverse Dirac operators on Riemannian foliations

We obtain a vanishing theorem for the half-kernel of a transverse Spin ^c Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle, whose curvature vanishes along the leaves and is transversely non-degenerate at any point of the ambient manifold.      

Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ?, there exists one more foliation (M, \(\mathcal{O}\)), which is generally singular and is called an aureole foliation; moreover, the foliations ? and \(\mathcal{O}\) have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ?/? with a compactly embedded Lie subalgebra ? in ?, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ?). We prove that for such foliations, there exists a unique minimal set ?, and ? is contained in the closure of any leaf. If the foliation (M, ?) is proper, then ? is a unique closed leaf of this foliation.     

Volume-Minimizing Foliations on Spheres

The volume of a k-dimensional foliation in a Riemannian manifold Mⁿ is defined as the mass of the image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller (Comment. Math. Helv. 61 (1986), 177–192), ‘singular’ foliations by 3-spheres are constructed on round spheres S⁴ⁿ⁺³, as well as a singular foliation by 7-spheres on S¹⁵, which minimize volume within their respective relative homology classes. These singular examples, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of S⁴ⁿ⁺³ and regular seven-dimensional foliations of S¹⁵, since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.The second author was supported during this research by grants from the Universidade de Sāo Paulo, FAPESP Proc. 1999/02684-5, and Lehigh University, and thanks those institutions for enabling the collaboration involved in this work.Mathematics Subject Classifications (2000). 53C12, 53C38.     

Spectrum Problems for Singular Integral Operators with Carleman Shift

In this paper we are concerned with the complete spectral analysis for the operator 𝒯 = 𝒳𝒮𝒰 in the space L_p(𝕋) (𝕋 denoting the unit circle), where 𝒳 is the characteristic function of some arc of 𝕋, 𝒮 is the singular integral operator with Cauchy kernel and 𝒰 is a Carleman shift operator which satisfies the relations 𝒰² = I and 𝒮𝒰 = ±𝒰𝒮, where the sign + or — is taken in dependence on whether 𝒰 is a shift operator on L_p(𝕋) preserving or changing the orientation of 𝕋. This includes the identification of the Fredholm and essential parts of the spectrum of the operator 𝒯, the determination of the defect numbers of 𝒯 — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.