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On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

Let ( M , g ) be a compact Riemannian manifold and P g an elliptic, formally self-adjoint, conformally covariant operator of order m acting on smooth sections of a bundle over M . We prove that if P g has no rigid eigenspaces (see Definition 2.2), the set of functions f ∈ C ∞ ( M , ℝ ) for which P e f g has only simple non-zero eigenvalues is a residual set in C ∞ ( M , ℝ ) . As a consequence we prove that if P g has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the C ∞ -topology....

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

On the relation between elliptic and parabolic Harnack inequalities

Waldemar Hebisch, Laurent Saloff-Coste (2001)

Annales de l’institut Fourier

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suffices to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suffices to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for Δ on M , (i.e., for ∂ t + Δ ) and elliptic Harnack inequality for - ∂ t 2 + Δ on ℝ × M .

On the Schrödinger heat kernel in horn-shaped domains

Gabriele Grillo (2004)

Colloquium Mathematicae

We prove pointwise lower bounds for the heat kernel of Schrödinger semigroups on Euclidean domains under Dirichlet boundary conditions. The bounds take into account non-Gaussian corrections for the kernel due to the geometry of the domain. The results are applied to prove a general lower bound for the Schrödinger heat kernel in horn-shaped domains without assuming intrinsic ultracontractivity for the free heat semigroup.

On the spectral theory and dynamics of asymptotically hyperbolic manifolds

Julie Rowlett (2010)

Annales de l’institut Fourier

We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.

On the spectrum of Riemannian submersions with totally geodesic fibers

Gérard Besson, Manlio Bordoni (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we give a rule to compute explicitely the spectrum and the eigenfunctions of the total space of a Riemannian submersion with totally geodesic fibers, in terms of the spectra and eigenfunctions of the typical fiber and any associated principal bundle.

Currently displaying 101 – 120 of 148