Asymptotic stability of the cross curvature flow at a hyperbolic metric

  title={Asymptotic stability of the cross curvature flow at a hyperbolic metric},
  author={Dan Knopf and Andrea Young},
We show that for any hyperbolic metric on a closed 3-manifold, there exists a neighborhood such that every solution of a normalized cross curvature flow with initial data in this neighborhood exists for all time and converges to a constant-curvature metric. We demonstrate that the same technique proves an analogous result for Ricci flow. Additionally, we prove short-time existence and uniqueness of cross curvature flow under slightly weaker regularity hypotheses than were previously known. 
Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvature
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence
Stability of complex hyperbolic space under curvature-normalized Ricci flow
Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two
The eternal solution to the cross curvature flow exists in 3-manifolds of negative sectional curvature
Given a closed 3-manifold M endowed with a radial symmetric metric of negative sectional curvature, we define the cross curvature flow on M ; using the maximum principle theorem, we demonstrated that
Generalized cross curvature flow
In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature
Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature
We consider the expansion of co-compact convex hypersurfaces in Minkowski space by functions of their curvature, and prove under quite general conditions that solutions are asymptotic to the
Cross curvature flow on a negatively curved solid torus
The classic 2 ‐Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3‐manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric.
Convergence Stability for Ricci Flow
The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if
Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings in Minkowski Space and the Cross Curvature Flow
This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review
Stability of Ricci-flat Spaces and Singularities in 4d Ricci Flow
In this thesis, we describe some closely related results on Ricci curvature and Ricci flow that we obtained during the last couple of years. In Chapter 1, we discuss the formation of singularities in
Dynamical stability of algebraic Ricci solitons
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds.


The Cross Curvature Flow of 3-manifolds with Negative Sectional Curvature
We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional
Cross Curvature Flow on Locally Homogenous Three-manifolds (I)
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the
Stability of the Ricci flow at Ricci-flat metrics
If g is a metric whose Ricci flow g (t) converges, one may ask if the same is true for metrics g that are small perturbations of g. We use maximal regularity theory and center manifold analysis to
Examples for cross curvature flow on 3-manifolds
Recently, B. Chow and R.S. Hamilton [3] introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over
Pinching constants for hyperbolic manifolds
SummaryWe show in this paper that for everyn≧4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with
Ricci flow, Einstein metrics and space forms
The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian
Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows
In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John
On the topology of the space of negatively curved metrics
We show that the space of negatively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n\geq 10$, is highly non-connected.
Analytic Semigroups and Optimal Regularity in Parabolic Problems
Introduction.- 0 Preliminary material: spaces of continuous and Holder continuous functions.- 1 Interpolation theory.- Analytic semigroups and intermediate spaces.- 3 Generation of analytic
Short-time existence of solutions to the cross curvature flow on 3-manifolds
This research was partially supported by an Australian Research Council Discovery grant entitled Geometric evolution equations and global effects of curvature.