2,181 research outputs found

### A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

We consider a subset $S$ of the complex Lie algebra \so(n,\C) and the cone
$C(S)$ of curvature operators which are nonnegative on $S$. We show that $C(S)$
defines a Ricci flow invariant curvature condition if $S$ is invariant under
\Ad_{\SO(n,\C)}. The analogue for K\"ahler curvature operators holds as well.
Although the proof is very simple and short it recovers all previously known
invariant nonnegativity conditions. As an application we reprove that a compact
K\"ahler manifold with positive orthogonal bisectional curvature evolves to a
manifold with positive bisectional curvature and is thus biholomorphic to
\CP^n. Moreover, the methods can also be applied to prove Harnack
inequalities. In addition to an earlier version the paper contains some remarks
on negative results for Harnack inequalities

### A duality theorem for Riemannian foliations in nonnegative sectional curvature

Using a new type of Jacobi field estimate we will prove a duality theorem for
singular Riemannian foliations in complete manifolds of nonnegative sectional
curvature

### Nonnegatively and Positively curved manifolds

This is a survey written for a special edition of the journal of differential
geometry

### Recent Results from the KTeV Experiment

We present recent preliminary results from five decay channels. From the
$K_L\to\pi^+\pi^-\gamma$ channel, we extract form factors for the CP violating
M1 direct photon emission amplitude and the fraction of the total decay
amplitude that is due to direct emission. We have placed an upper limit on the
$K_L\to\pi^0\pi^0\gamma$ branching ratio, and preliminary measurements of the
$K_L\to\pi^{\pm}e^{\mp}\nu e^+e^-$ and $\pi^0\to e^+e^-$ branching ratios are
presented. Finally, we report measurements of both the branching ratio and the
form factor parameters for the decay $K_L\to e^+e^-\gamma$.Comment: Proceedings from Rencontres De Moriond 2006: Electroweak Interactions
and Unified Theorie

### 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 $n$-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.

### Riemannian foliations of spheres

We show that a Riemannian foliation on a topological $n$-sphere has leaf
dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the
fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to
classify Riemannian foliations on round spheres up to metric congruence.Comment: 17 page

### Manifolds with positive curvature operators are space forms

We confirm a conjecture of Hamilton: On compact manifolds the normalized
Ricci flow evolves metrics with positive curvature operators to limit metrics
with constant curvature

### On the Berger conjecture for manifolds all of whose geodesics are closed

A conjecture of Berger states that, for any simply connected Riemannian
manifold all of whose geodesics are closed, all prime geodesics have the same
length. We firstly show that the energy function on the free loop space of such
a manifold is a perfect Morse-Bott function with respect to a suitable
cohomology. Secondly we explain when the negative bundles along the critical
manifolds are orientable. These two general results then lead to a solution of
Berger's conjecture when the underlying manifold is a sphere of dimension at
least four.Comment: 39 page

### Revisiting homogeneous spaces with positive curvature

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard
Bergery's classification of odd dimensional positively curved homogeneous
spaces. Since this classification has been used in other papers as well, we
give a modern, complete and self contained proof (in odd as well as even
dimensions), confirming that there are indeed no new examples.Comment: 13 page

### How to produce a Ricci Flow via Cheeger-Gromoll exhaustion

We prove short time existence for the Ricci flow on open manifolds of
nonnegative complex sectional curvature. We do not require upper curvature
bounds. By considering the doubling of convex sets contained in a
Cheeger-Gromoll convex exhaustion and solving the singular initial value
problem for the Ricci flow on these closed manifolds, we obtain a sequence of
closed solutions of the Ricci flow with nonnegative complex sectional curvature
which subconverge to a solution of the Ricci flow on the open manifold.
Furthermore, we find an optimal volume growth condition which guarantees long
time existence, and we give an analysis of the long time behaviour of the Ricci
flow. Finally, we construct an explicit example of an immortal nonnegatively
curved solution of the Ricci flow with unbounded curvature for all time.Comment: 42 pages, Added: a) example of an immortal solution which flows from
bounded to unbounded curvature in finite time, b) long time analysis of the
Ricci flo

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