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

TL;DR

This paper proves the long-term existence and convergence of the cross curvature flow on closed 3-manifolds with negative sectional curvature, showing it evolves metrics towards hyperbolic geometry.

Contribution

It establishes the long-time existence and convergence of the cross curvature flow on negatively curved 3-manifolds, a significant step in geometric analysis.

Findings

Flow exists for all time on negatively curved 3-manifolds

Flow converges to a hyperbolic metric

Solution remains smooth throughout the evolution

Abstract

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 of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric