2,249 research outputs found
Scalar Curvature and Intrinsic Flat Convergence
Herein we present open problems and survey examples and theorems concerning
sequences of Riemannian manifolds with uniform lower bounds on scalar curvature
and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought
to have certain limit spaces do not converge with respect to smooth or
Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic
Flat convergence, developed jointly with Wenger. This notion has been applied
successfully to study sequences that arise in General Relativity. Gromov has
suggested it should be applied in other settings as well. We first review
intrinsic flat convergence, its properties, and its compactness theorems,
before presenting the applications and the open problems.Comment: 53 pages, 4 figures, Geometric Analysis on Riemannian and singular
metric spaces, Lake Como School of Advnaced Studies, 11-15 July 2016, v2:
minor fixes as requested by referee, to appear as a chapter in the deGruyter
book series on PDE and Measure Theory edited by Gigl
Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups
In 1968, Milnor conjectured that a complete noncompact manifold with
nonnegative Ricci curvature has a finitely generated fundamental group. The
author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two
theorems. The first states that if such a manifold has small linear diameter
growth then its fundamental group is finitely generated. The second states that
if such a manifold has an infinitely generated fundamental group then it has a
tangent cone at infinity which is not polar. A corollary of either theorem is
the fact that if such a manifold has linear volume growth, then its fundamental
group is finitely generated.Comment: 12 pages, figures available from author [email protected]
Friedmann Cosmology and Almost Isotropy
In the Friedmann Model of the universe, cosmologists assume that spacelike
slices of the universe are Riemannian manifolds of constant sectional
curvature. This assumption is justified via Schur's Theorem by stating that the
spacelike universe is locally isotropic. Here we define a Riemannian manifold
as almost locally isotropic in a sense which allows both weak gravitational
lensing in all directions and strong gravitational lensing in localized angular
regions at most points. We then prove that such a manifold is Gromov Hausdorff
close to a length space which is a collection of space forms joined at
discrete points. Within the paper we define a concept we call an "exponential
length space" and prove that if such a space is locally isotropic then it is a
space form.Comment: 45 pages, 7 figures (eps), AIM Workshop on General Relativity 2002,
added some details to the explanation, fixed typos/gramma
How Riemannian Manifolds Converge: A Survey
This is an intuitive survey of extrinsic and intrinsic notions of convergence
of manifolds complete with pictures of key examples and a discussion of the
properties associated with each notion. We begin with a description of three
extrinsic notions which have been applied to study sequences of submanifolds in
Euclidean space: Hausdorff convergence of sets, flat convergence of integral
currents, and weak convergence of varifolds. We next describe a variety of
intrinsic notions of convergence which have been applied to study sequences of
compact Riemannian manifolds: Gromov-Hausdorff convergence of metric spaces,
convergence of metric measure spaces, Instrinsic Flat convergence of integral
current spaces, and ultralimits of metric spaces. We close with a speculative
section addressing possible notions of intrinsic varifold convergence,
convergence of Lorentzian manifolds and area convergence.Comment: Solicited survey article for a volume of articles in honor of
Cheeger's 65th Birthday. Version 2 adds additional references concerning
convergence of Lorentzian manifolds and updates citation
The Almost Rigidity of Manifolds with Lower Bounds on Ricci Curvature and Minimal Volume Growth
We consider complete noncompact Riemannian manifolds with quadratically
decaying lower Ricci curvature bounds and minimal volume growth. We first prove
a rigidity result showing that ends with strongly minimal volume growth are
isometric to warped product manifolds. Next we consider the almost rigid case
in which manifolds with nonnegative and quadratically decaying lower Ricci
curvature bounds have minimal volume growth. Compact regions in such manifolds
are shown to be asymptotically close to warped products in the Gromov-Hausdorff
topology. Manifolds with nonnegative Ricci curvature and linear volume growth
are shown to have regions which are asymptotically close to being isometric
products. The proofs involve a careful analysis of the Busemann functions on
these manifolds using the recently developed Cheeger-Colding Almost Rigidity
Theory. In addition, we show that the diameters of the level sets of Busemann
functions in such manifolds grow sublinearly.Comment: To appear in: Communications in Analysis and Geometry, 54pp.
(submission date to CAG: Dec 1997
Oberwolfach Report: Spacetime Intrinsic Flat Convergence
This Oberwolfach report describes published work with Carlos Vega introducing
the null distance on a spacetime, and announces unpublished applications of
this work jointly with Carlos Vega and with Anna Sakovich applying the null
distance to define a spacetime intrinsic flat convergence (SF convergence). The
announced work with Vega concerns convergence of sequences of big bang
spacetimes and the announced work with Anna Sakovich concerns convergence of
future maximal developments of initial data sets. Both were originally
announced at the 2018 JMM. This report lists some open questions that will be
discussed at the Oberwolfach meeting this coming August.Comment: 3 page
The Tetrahedral Property and a new Gromov-Hausdorff Compactness Theorem
We present the Tetrahedral Compactness Theorem which states that sequences of
Riemannian manifolds with a uniform upper bound on volume and diameter that
satisfy a uniform tetrahedral property have a subsequence which converges in
the Gromov-Hausdorff sense to a countably rectifiable metric
space of the same dimension. The tetrahedral property depends only on distances
between points in spheres, yet we show it provides a lower bound on the volumes
of balls. The proof is based upon intrinsic flat convergence and a new notion
called the sliced filling volume of a ball.Comment: This is an announcement with 1 figur
Universal Covers for Hausdorff Limits of Noncompact Spaces
We prove that if is the Gromov-Hausdorff limit of a sequence of complete
manifolds, , with a uniform lower bound on Ricci curvature then has
a universal cover.Comment: 34 page
Relating Notions of Convergence in Geometric Analysis
We relate convergence of metric tensors or volume convergence to a
given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for
sequences of Riemannian manifolds. We present many examples of sequences of
conformal metrics which demonstrate that these notions of convergence do not
agree in general even when the sequence is conformal, , to a
fixed manifold. We then prove a theorem demonstrating that when sequences of
metric tensors on a fixed manifold are bounded, , and either the volumes converge, , or the metric tensors converge in the sense,
then the Riemannian manifolds converge in the measured
Gromov-Hausdorff and volume preserving Intrinsic Flat sense to .Comment: 41 pages, 3 figures v2: Updated proof of Theorem 4.4 v3: Three
figures added and referee comments addressed. To appear in Nonlinear Analysi
Hausdorff Convergence and Universal Covers
We prove that if is the Gromov-Hausdorff limit of a sequence of compact
manifolds, , with a uniform lower bound on Ricci curvature and a uniform
upper bound on diameter, then has a universal cover. We then show that, for
sufficiently large, the fundamental group of has a surjective
homeomorphism onto the group of deck transforms of . Finally, in the
non-collapsed case where the have an additional uniform lower bound on
volume, we prove that the kernels of these surjective maps are finite with a
uniform bound on their cardinality. A number of theorems are also proven
concerning the limits of covering spaces and their deck transforms when the
are only assumed to be compact length spaces with a uniform upper bound
on diameter.Comment: 17 Page
- β¦