Mean curvature flow in a Ricci flow background

Details Mean curvature flow in a Ricci flow background Mean curvature flow in a Ricci flow background

2011-05-30

arxiv.org/abs/1105.6081v3

Mathematics

Differential Geometry

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Scientific paper

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.

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