# Asymptotic stability of the cross curvature flow at a hyperbolic metric

@inproceedings{Knopf2006AsymptoticSO, title={Asymptotic stability of the cross curvature flow at a hyperbolic metric}, author={Dan Knopf and Andrea Young}, year={2006} }

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.

## 23 Citations

Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvature

- Mathematics
- 2016

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

- Mathematics
- 2011

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

- MathematicsTURKISH JOURNAL OF MATHEMATICS
- 2019

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

- Mathematics
- 2021

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

- Mathematics
- 2015

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

- Mathematics
- 2010

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

- MathematicsThe Journal of Geometric Analysis
- 2019

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

- Mathematics
- 2019

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

- Mathematics
- 2012

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

- Mathematics
- 2016

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.…

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This research was partially supported by an Australian Research Council Discovery grant
entitled Geometric evolution equations and global effects of curvature.