215 research outputs found
A Conformally Invariant Holographic Two-Point Function on the Berger Sphere
We apply our previous work on Green's functions for the four-dimensional
quaternionic Taub-NUT manifold to obtain a scalar two-point function on the
homogeneously squashed three-sphere (otherwise known as the Berger sphere),
which lies at its conformal infinity. Using basic notions from conformal
geometry and the theory of boundary value problems, in particular the
Dirichlet-to-Robin operator, we establish that our two-point correlation
function is conformally invariant and corresponds to a boundary operator of
conformal dimension one. It is plausible that the methods we use could have
more general applications in an AdS/CFT context.Comment: 1+49 pages, no figures. v2: Several typos correcte
Minimal energy problems for strongly singular Riesz kernels
We study minimal energy problems for strongly singular Riesz kernels on a
manifold. Based on the spatial energy of harmonic double layer potentials, we
are motivated to formulate the natural regularization of such problems by
switching to Hadamard's partie finie integral operator which defines a strongly
elliptic pseudodifferential operator on the manifold. The measures with finite
energy are shown to be elements from the corresponding Sobolev space, and the
associated minimal energy problem admits a unique solution. We relate our
continuous approach also to the discrete one, which has been worked out earlier
by D.P. Hardin and E.B. Saff.Comment: 31 pages, 2 figure
Diffractive Theorems for the Wave Equation with Inverse Square Potential
We first establish the presence of a diffractive front in the fundamental
solution of the wave operator with a diract delta intial condition in two
dimensional euclidean space caused by the potentials perturbation on the
spherical laplacian. This motivates a result which restricts the propagation of
singularities for the wave operator with a more general potential to precisely
these diffractive fronts higher dimensional euclidean spaces. This is proven
using microlocal energy estimates.Comment: 41 pages, 6 figure
Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes
We establish a Bohr-Sommerfeld type condition for quasi-normal modes of a
slowly rotating Kerr-de Sitter black hole, providing their full asymptotic
description in any strip of fixed width. In particular, we observe a
Zeeman-like splitting of the high multiplicity modes at a=0 (Schwarzschild-de
Sitter), once spherical symmetry is broken. The numerical results presented in
Appendix B show that the asymptotics are in fact accurate at very low energies
and agree with the numerical results established by other methods in the
physics literature. We also prove that solutions of the wave equation can be
asymptotically expanded in terms of quasi-normal modes; this confirms the
validity of the interpretation of their real parts as frequencies of
oscillations, and imaginary parts as decay rates of gravitational waves.Comment: 66 pages, 6 figures; journal version (to appear in Annales Henri
Poincar\'e
Index in K-theory for families of fibred cusp operators
A families index theorem in K-theory is given for the setting of Atiyah,
Patodi and Singer of a family of Dirac operators with spectral boundary
condition. This result is deduced from such a K-theory index theorem for the
calculus of cusp, or more generally fibred cusp, pseudodifferential operators
on the fibres (with boundary) of a fibration; a version of Poincare duality is
also shown in this setting, identifying the stable Fredholm families with
elements of a bivariant K-group.Comment: 64 pages, corrected typo
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