Algebraic & Geometric Topology

Cross curvature flow on a negatively curved solid torus

Jason DeBlois, Dan Knopf, and Andrea Young

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The classic 2π–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. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the “2π–metric” and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds and integral convergence to hyperbolic for the metrics under consideration.

Article information

Algebr. Geom. Topol., Volume 10, Number 1 (2010), 343-372.

Received: 24 June 2009
Revised: 25 November 2009
Accepted: 17 December 2009
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 57M50: Geometric structures on low-dimensional manifolds 58J35: Heat and other parabolic equation methods 58J32: Boundary value problems on manifolds

cross curvature flow 2$\pi$–theorem


DeBlois, Jason; Knopf, Dan; Young, Andrea. Cross curvature flow on a negatively curved solid torus. Algebr. Geom. Topol. 10 (2010), no. 1, 343--372. doi:10.2140/agt.2010.10.343.

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