1,196 research outputs found
Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
We prove a new kind of estimate that holds on any manifold with lower Ricci
bounds. It relates the geometry of two small balls with the same radius,
potentially far apart, but centered in the interior of a common minimizing
geodesic. It reveals new, previously unknown, properties that all generalized
spaces with a lower Ricci curvature bound must have and it has a number of
applications.
This new kind of estimate asserts that the geometry of small balls along any
minimizing geodesic changes in a H\"older continuous way with a constant
depending on the lower bound for the Ricci curvature, the dimension of the
manifold, and the distance to the end points of the geodesic. We give examples
that show that the H\"older exponent, along with essentially all the other
consequences that we show follow from this estimate, are sharp. The unified
theme for all of these applications is convexity.
Among the applications is that the regular set is convex for any
non-collapsed limit of Einstein metrics. In the general case of potentially
collapsed limits of manifolds with just a lower Ricci curvature bound we show
that the regular set is weakly convex and convex, that is almost every
pair of points can be connected by a minimizing geodesic whose interior is
contained in the regular set. We also show two conjectures of Cheeger-Colding.
One of these asserts that the isometry group of any, even collapsed, limit of
manifolds with a uniform lower Ricci curvature bound is a Lie group; the key
point for this is to rule out small subgroups. The other asserts that the
dimension of any limit space is the same everywhere. Finally, we show that a
Reifenberg type property holds for collapsed limits and discuss why this
indicate further regularity of manifolds and spaces with Ricci curvature
bounds.Comment: 48 page
Inverse limit spaces satisfying a Poincare inequality
We give conditions on Gromov-Hausdorff convergent inverse systems of metric
measure graphs (and certain higher dimensional inverse systems of metric
measure spaces) which imply that the measured Gromov-Hausdorff limit
(equivalently, the inverse limit) is a PI space, i.e. it satisfies a doubling
condition and a Poincare inequality in the sense of Heinonen-Koskela. We also
give a systematic construction of examples for which our conditions are
satisfied. Included are known examples of PI spaces, such as Laakso spaces, and
a large class of new examples. Generically our graph examples have the property
that they do not bilipschitz embed in any Banach space with Radon-Nikodym
property, but they do embed in the Banach space L_1. For Laakso spaces, these
facts were discussed in our earlier papers
Characterization of the Radon-Nikodym Property in terms of inverse limits
We clarify the relation between inverse systems, the Radon-Nikodym property,
the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in
our earlier paper on differentiability of Lipschitz maps into Banach spaces
Regularity of Einstein Manifolds and the Codimension 4 Conjecture
In this paper, we are concerned with the regularity of noncollapsed
Riemannian manifolds with bounded Ricci curvature, as well as their
Gromov-Hausdorff limit spaces , where denotes the Riemannian distance. Our main result is a
solution to the codimension conjecture, namely that is smooth away from
a closed subset of codimension . We combine this result with the ideas of
quantitative stratification to prove a priori estimates on the full
curvature for all . In the case of Einstein manifolds, we improve
this to estimates on the regularity scale. We apply this to prove a conjecture
of Anderson that the collection of -manifolds with
, , and contains at most a
finite number of diffeomorphism classes. A local version of this is used to
show that noncollapsed -manifolds with bounded Ricci curvature have a priori
Riemannian curvature estimates.Comment: Estimates in Theorem 1.9 shown to hold in the distribution sense; so
interpreted in Definition 1.1
A weakly second order differential structure on rectifiable metric measure spaces
We give the definition of angles on a Gromov-Hausdorff limit space of a
sequence of complete n-dimensional Riemannian manifolds with a lower Ricci
curvature bound. We apply this to prove there is a weakly second order
differential structure on these spaces and prove there is a unique Levi-Civita
connection allowing us to define the Hessian of a twice differentiable
function.Comment: 29 pages. Several statements added. The title change
Compression bounds for Lipschitz maps from the Heisenberg group to
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg
group with its Carnot-Carath\'eodory metric and apply it to give a lower bound
on the integrality gap of the Goemans-Linial semidefinite relaxation of the
Sparsest Cut problem
Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates
Consider a sequence of pointed n-dimensional complete Riemannian manifolds
{(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and
g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard
Hamilton showed that if the initial injectivity radii are uniformly bounded
below then there is a subsequence which converges to an n-dimensional solution
to the Ricci flow. We prove a generalization of this theorem where the initial
metrics may collapse. Without injectivity radius bounds we must allow for
convergence in the Gromov-Hausdorff sense to a space which is not a manifold
but only a metric space. We then look at the local geometry of the limit to
understand how it relates to the Ricci flow.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper14.abs.htm
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