Journals / Turkish Journal of Mathematics / 2004 / Cilt: 28 - Sayı: 1
The Cross Curvature Flow of 3-Manifolds with Negative Sectional Curvature
- Pages
- 1–10
- DOI
- —
Abstract
We consider the cross curvature flow, an evolution equation of metrics on 3-manifolds. We establish short time existence when the sectional curvature has a sign. In the case of negative sectional curvature, we obtain some monotonicity formulas which support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time.
Özet
We consider the cross curvature flow, an evolution equation of metrics on 3-manifolds. We establish short time existence when the sectional curvature has a sign. In the case of negative sectional curvature, we obtain some monotonicity formulas which support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time.