The gradient flow for renormalized volume

Ken Bromberg (Utah)

Abstract: If a 3-manifold supports a hyperbolic metric of infinite volume it will support a large family of such metrics. Based on ideas from physics due to Graham and Witten, Krasnov and Schlenker defined the notion of the “renormalized volume” of a hyperbolic 3-manifold. The renormalized volume assigns a finite volume to a hyperbolic manifold that would have infinite volume in the usual sense. This is a smooth function on the space of hyperbolic metrics and furthermore there is a simple formula (due to Krasnov and Schlenker) for its differential. We will describe joint work with M. Bridgeman and J. Brock on the gradient flow of renormalized volume and some consequences for the geometry of the hyperbolic manifolds. Finally we will speculate on how renormalized volume may give a new approach to the hyperbolization theorem for 3-manifolds.