59 research outputs found
Curvature bounds via Ricci smoothing
We give a proof of the fact that the upper and the lower sectional curvature
bounds of a complete manifold vary at a bounded rate under the Ricci flow
Regularity of limits of noncollapsing sequences of manifolds
We prove that iterated spaces of directions of a limit of a noncollapsing
sequence of manifolds with lower curvature bound are topologically spheres. As
an application we show that for any finite dimensional Alexandrov space
with there exists an Alexandrov space homeomorphic to
which can not be obtained as such a limit.Comment: 17 pages, to appear in GAF
Perelman's Stability Theorem
We give a proof of the celebrated stability theorem of Perelman stating that
for a noncollapsing sequence of Alexandrov spaces with curvature bounded
below Gromov-Hausdorff converging to a compact Alexandrov space , is
homeomorphic to for all large .Comment: revised introductio
Restrictions on collapsing with a lower sectional curvature bound
We obtain new topological information about the local structure of collapsing
under a lower sectional curvature bound. As an application we prove a new
sphere theorem and obtain a partial result towards the conjecture that not
every Alexandrov space can be obtained as a limit of a sequence of Riemannian
manifolds with sectional curvature bounded from below
Structure of fundamental groups of manifolds with Ricci curvature bounded below
Verifying a conjecture of Gromov we establish a generalized Margulis Lemma
for manifolds with lower Ricci curvature bound. Among the various applications
are finiteness results for fundamental groups of compact -manifolds with
upper diameter and lower Ricci curvature bound modulo nilpotent normal
subgroups.Comment: 49 p.,some typos removed, more details in section 1 and in the proof
of Lemma 3.
On noncollapsed almost Ricci-flat 4-manifolds
We give topological conditions to ensure that a noncollapsed almost
Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is
that the manifold is spin and has a nonzero A-hat genus. Another condition is
that the fundamental group is infinite or, more generally, of sufficiently
large cardinality.Comment: 21 pages, final versio
Weakly noncollapsed RCD spaces with upper curvature bounds
We show that if a space with has
curvature bounded from above by in the sense of Alexandrov then
.Comment: 14 page
Finiteness theorems for nonnegatively curved vector bundles
We prove several finiteness theorems for the normal bundles to souls in
nonnegatively curved manifolds. More generally, we obtain finiteness results
for open Riemannian manifolds whose topology is concentrated on compact domains
of ``bounded geometry''.Comment: 26 page
CD meets CAT
We show that if a noncollapsed space with has
curvature bounded above by in the sense of Alexandrov then and is an Alexandrov space of curvature bounded below by
. We also show that if a space with finite
has curvature bounded above then it is infinitesimally Hilbertian.Comment: We add a new section where we prove a new theorem that if a
space with finite has curvature bounded above then it is infinitesimally
Hilbertian. Using this we remove the infinitesimal Hilbertianness assumption
from the main theorem. Minor corrections, additional reference
Obstructions to nonnegative curvature and rational homotopy theory
We establish a link between rational homotopy theory and the problem which
vector bundles admit complete Riemannian metric of nonnegative sectional
curvature. As an application, we show for a large class of simply-connected
nonnegatively curved manifolds that, if C lies in the class and T is a torus of
positive dimension, then "most" vector bundles over the product of C and T
admit no complete nonnegatively curved metric.Comment: Fixed an error is the proof of lemma B.
- …