The eternal solution to the cross curvature flow exists in 3-manifolds of negative sectional curvature

Abstract

Given a closed 3-manifold $M^3$ endowed with a radial symmetric metric of negative sectional curvature, wedefine the cross curvature flow on $M^3$; using the maximum principle theorem, we demonstrated that the solution to thecross curvature flow exists for all time and converges pointwise to a hyperbolic metric.