675 research outputs found
Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes
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
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
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
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
Diffeomorphism Invariant Integrable Field Theories and Hypersurface Motions in Riemannian Manifolds
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
at which the hypersurface passes the point . 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
-branesmoving in an arbitrary -dimensional Riemannian manifold, the
models are integrable: In the time-function formulation the equation becomes
linear (with 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, -component theories possessing infinitely many
diffeomorphism invariant, Poisson commuting, conserved charges.Comment: 15 pages, LATE
- …