675 research outputs found

    Deforming hypersurfaces of the sphere by their mean-curvature

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    Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes

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    Spacelike hypersurfaces of prescribed mean curvature in cosmological space times are constructed as asymptotic limits of a geometric evolution equation. In particular, an alternative, constructive proof is given for the existence of maximal and constant mean curvature slices

    Convex ancient solutions of the mean curvature flow

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    We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere

    Ancient solutions to the Ricci flow with pinched curvature

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    We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.Comment: to appear in Duke Math Journa

    Mean curvature flow in a Ricci flow background

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    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.Comment: final versio

    Capillary surfaces over obstacles

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    Contracting convex hypersurfaces in Riemannian-manifolds by their mean-curvature

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    Diffeomorphism Invariant Integrable Field Theories and Hypersurface Motions in Riemannian Manifolds

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    We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function τ(x)\tau(x) at which the hypersurface passes the point xx. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism invariant field theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean MM-branesmoving in an arbitrary M+1M+1-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear (with τ(x)\tau(x) a harmonic function on the embedding Riemannian manifold). We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in \Real^3 by methods used in electrostatics and point out an additional gradient flow structure in \Real^n. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, NN-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges.Comment: 15 pages, LATE
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