23,059 research outputs found
Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case
This is the first paper of a series in which we plan to study spectral
asymptotics for sub-Riemannian Laplacians and to extend results that are
classical in the Riemannian case concerning Weyl measures, quantum limits,
quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators
have been well studied from the point of view of PDEs, global geometrical and
dynamical aspects have not been the subject of much attention. As we will see,
already in the simplest case, the statements of the results in the
sub-Riemannian setting are quite different from those in the Riemannian one.
Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional
manifold with an oriented contact distribution. There exists a privileged
choice of the contact form, with an associated Reeb vector field and a
canonical volume form that coincides with the Popp measure. We establish a
Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR
Laplacian under the assumption that the Reeb flow is ergodic. The limit measure
is given by the normalized Popp measure.This is the first time that such a
result is established for a hypoelliptic operator, whereas the usual Shnirelman
theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian
manifold with ergodic geodesic flow.To prove our theorem, we first establish a
microlocal Weyl law, which allows us to identify the limit measure and to prove
the microlocal concentration of the eigenfunctions on the characteristic
manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this
characteristic manifold, thus showing that, in some sense, all 3D contact
structures are microlocally equivalent. The quantum version of this normal form
provides a useful microlocal factorization of the sR Laplacian. Using the
normal form, the factorization and the ergodicity assumption, we finally
establish a variance estimate, from which QE follows.We also obtain a second
result, which is valid without any ergodicity assumption: every Quantum Limit
(QL) can be decomposed in a sum of two mutually singular measures: the first
measure is supported on the unit cotangent bundle and is invariant under the sR
geodesic flow, and the second measure is supported on the characteristic
manifold of the sR Laplacian and is invariant under the lift of the Reeb flow.
Moreover, we prove that the first measure is zero for most QLs.Comment: to appear in Duke Math.
The structure and interpretation of cosmology: Part II - The concept of creation in inflation and quantum cosmology
The purpose of the paper, of which this is part II, is to review, clarify,
and critically analyse modern mathematical cosmology. The emphasis is upon
mathematical objects and structures, rather than numerical computations. Part
II provides a critical analysis of inflationary cosmology and quantum
cosmology, with particular attention to the claims made that these theories can
explain the creation of the universe
Conformally Flat Circle Bundles over Surfaces
We classify conformally flat Riemannian manifolds which possesses a free
isometric action.Comment: 12 pages, Part of the author's PhD thesi
Global Aspects of Abelian Duality in Dimension Three
In three dimensions, an abelian gauge field is related by duality to a free,
periodic scalar field. Though usually considered on Euclidean space, this
duality can be extended to a general three-manifold M, in which case
topological features of M become important. Here I comment upon several of
these features as related to the partition function on M. In a companion
article, arXiv:1405.2483, I discuss similarly the algebra of operators on a
surface of genus g.Comment: 62 pages, 1 figure, v2: references adde
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