Mean Curvature Flow in a Ricci Flow Background
نویسندگان
چکیده
Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-withboundary. 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 differential 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.
منابع مشابه
Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow
Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...
متن کاملMean Curvature Driven Ricci Flow
We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.
متن کاملLectures on Mean Curvature Flow (MAT 1063 HS)
The mean curvature flow arises material science and condensed matter physics and has been recently successfully applied by Huisken and Sinestrari to topological classification of surfaces and submanifolds. It is closely related to the Ricci and inverse mean curvature flow. The most interesting aspect of the mean curvature flow is formation of singularities, which is the main theme of these lect...
متن کاملLectures on Mean Curvature Flow and Stability (MAT 1063 HS)
The mean curvature flow (MCF) arises material science and condensed matter physics and has been recently successfully applied to topological classification of surfaces and submanifolds. It is closely related to the Ricci and inverse mean curvature flow. The most interesting aspect of the mean curvature flow is formation of singularities, which is the main theme of these lectures. In dealing wit...
متن کاملRicci Flow and Nonnegativity of Sectional Curvature
In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of t...
متن کامل