619 research outputs found
Convergence of vector bundles with metrics of Sasaki-type
If a sequence of Riemannian manifolds, , converges in the pointed
Gromov-Hausdorff sense to a limit space, , and if are vector
bundles over endowed with metrics of Sasaki-type with a uniform upper
bound on rank, then a subsequence of the converges in the pointed
Gromov-Hausdorff sense to a metric space, . The projection maps
converge to a limit submetry and the fibers converge to
its fibers; the latter may no longer be vector spaces but are homeomorphic to
, where is a closed subgroup of ---called the {\em wane
group}--- that depends on the basepoint and that is defined using the holonomy
groups on the vector bundles. The norms converges to a map
compatible with the re-scaling in and the -action
on converges to an action on compatible with the
limiting norm.
In the special case when the sequence of vector bundles has a uniform lower
bound on holonomy radius (as in a sequence of collapsing flat tori to a
circle), the limit fibers are vector spaces. Under the opposite extreme, e.g.
when a single compact -dimensional manifold is re-scaled to a point, the
limit fiber is where is the closure of the holonomy group of the
compact manifold considered.
An appropriate notion of parallelism is given to the limiting spaces by
considering curves whose length is unchanged under the projection. The class of
such curves is invariant under the -action and each such curve preserves
norms. The existence of parallel translation along rectifiable curves with
arbitrary initial conditions is also exhibited. Uniqueness is not true in
general, but a necessary condition is given in terms of the aforementioned wane
groups .Comment: 44 pages, 1 figure, in V.2 added Theorem E and Section 4 on
parallelism in the limit space
The fundamental solution and Strichartz estimates for the Schr\"odinger equation on flat euclidean cones
We study the Schr\"odinger equation on a flat euclidean cone of cross-sectional radius , developing
asymptotics for the fundamental solution both in the regime near the cone point
and at radial infinity. These asymptotic expansions remain uniform while
approaching the intersection of the "geometric front", the part of the solution
coming from formal application of the method of images, and the "diffractive
front" emerging from the cone tip. As an application, we prove Strichartz
estimates for the Schr\"odinger propagator on this class of cones.Comment: 21 pages, 4 figures. Minor typos corrected. To be published in Comm.
Math. Phy
Quantum scalar field in D-dimensional static black hole space-times
An Euclidean approach for investigating quantum aspects of a scalar field
living on a class of D-dimensional static black hole space-times, including the
extremal ones, is reviewed. The method makes use of a near horizon
approximation of the metric and -function formalism for evaluating the
partition function and the expectation value of the field fluctuations
. After a review of the non-extreme black hole case, the extreme
one is considered in some details. In this case, there is no conical
singularity, but the finite imaginary time compactification introduces a cusp
singularity. It is found that the -function regularized partition
function can be defined, and the quantum fluctuations are finite on the
horizon, as soon as the cusp singularity is absent, and the corresponding
temperature is T=0.Comment: 9 pages, LaTe
Deligne-Beilinson cohomology and abelian link invariants: torsion case
For the abelian Chern-Simons field theory, we consider the quantum functional
integration over the Deligne-Beilinson cohomology classes and present an
explicit path-integral non-perturbative computation of the Chern-Simons link
invariants in , a toy example of 3-manifold with
torsion
Foliations and Chern-Heinz inequalities
We extend the Chern-Heinz inequalities about mean curvature and scalar
curvature of graphs of -functions to leaves of transversally oriented
codimension one -foliations of Riemannian manifolds. That extends
partially Salavessa's work on mean curvature of graphs and generalize results
of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and
Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of
3-dimensional Riemannian manifolds by constant mean curvature surfaces. These
Chern-Heinz inequalities for foliations can be applied to prove
Haymann-Makai-Osserman inequality (lower bounds of the fundamental tones of
bounded open subsets in terms of its inradius)
for embedded tubular neighborhoods of simple curves of .Comment: This paper is an improvment of an earlier paper titled On Chern-Heinz
Inequalities. 8 Pages, Late
The ground state and the long-time evolution in the CMC Einstein flow
Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold M
with non-positive Yamabe invariant (Y(M)). As noted by Fischer and Moncrief,
the reduced volume V(k)=(-k/3)^{3}Vol_{g(k)}(M) is monotonically decreasing in
the expanding direction and bounded below by V_{\inf}=(-1/6)Y(M))^{3/2}.
Inspired by this fact we define the ground state of the manifold M as "the
limit" of any sequence of CMC states {(g_{i},K_{i})} satisfying: i. k_{i}=-3,
ii. V_{i} --> V_{inf}, iii. Q_{0}((g_{i},K_{i}))< L where Q_{0} is the
Bel-Robinson energy and L is any arbitrary positive constant. We prove that (as
a geometric state) the ground state is equivalent to the Thurston
geometrization of M. Ground states classify naturally into three types. We
provide examples for each class, including a new ground state (the Double Cusp)
that we analyze in detail. Finally consider a long time and cosmologically
normalized flow (\g,\K)(s)=((-k/3)^{2}g,(-k/3))K) where s=-ln(-k) is in
[a,\infty). We prove that if E_{1}=E_{1}((\g,\K))< L (where E_{1}=Q_{0}+Q_{1},
is the sum of the zero and first order Bel-Robinson energies) the flow
(\g,\K)(s) persistently geometrizes the three-manifold M and the geometrization
is the ground state if V --> V_{inf}.Comment: 40 pages. This article is an improved version of the second part of
the First Version of arXiv:0705.307
Maximum solutions of normalized Ricci flows on 4-manifolds
We consider maximum solution , , to the normalized
Ricci flow. Among other things, we prove that, if is a smooth
compact symplectic 4-manifold such that and let
, be a solution to (1.3) on whose Ricci curvature
satisfies that and additionally , then there exists an , and a sequence of points
, , satisfying that, by passing to a
subsequence, , in the -pointed
Gromov-Hausdorff sense for any sequence , where
, , are complete complex hyperbolic orbifolds
of complex dimension 2 with at most finitely many isolated orbifold points.
Moreover, the convergence is in the non-singular part of
and
, where
(resp. ) is the Euler characteristic (resp. signature) of
.Comment: 23 page
A compactness theorem for complete Ricci shrinkers
We prove precompactness in an orbifold Cheeger-Gromov sense of complete
gradient Ricci shrinkers with a lower bound on their entropy and a local
integral Riemann bound. We do not need any pointwise curvature assumptions,
volume or diameter bounds. In dimension four, under a technical assumption, we
can replace the local integral Riemann bound by an upper bound for the Euler
characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.Comment: 28 pages, final version, to appear in GAF
On Breakdown Criteria for Nonvacuum Einstein Equations
The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski
stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can
be further extended in time if the second fundamental form and the derivative
of the lapse of the foliation are uniformly bounded. This theorem and its proof
were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the
author's Ph.D. thesis. In this paper, we state the main results of the thesis,
and we summarize and discuss their proofs. In particular, we will discuss the
various issues resulting from nontrivial Ricci curvature and the coupling
between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6
(geometry of null cones
Heat kernels on curved cones
A functorial derivation is presented of a heat-kernel expansion coefficient
on a manifold with a singular fixed point set of codimension two. The existence
of an extrinsic curvature term is pointed out.Comment: 4p.,sign errors corrected and a small addition,uses JyTeX,MUTP/94/0
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