Gerhard Huisken

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

which shrink towards the center of the initial sphere in finite time. It was shown in [3], that this behaviour is very typical: If the initial hypersurface M o o R , + 1 is uniformly convex, then the surfaces M t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction. If the ambient space N is a general Riemannian manifold, the curvature of N will interfere with the mot ion of the surfaces M r We want to show here that the contract ion first to a small sphere and then to a single point is still

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

Spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes 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.

Evolution Equations in Geometry

Partial differential equations have been used for a long time to model the evolution of physical systems in time, the theory of their solutions has often been developed in close correspondence to a progressive understanding and continuing development of the underlying physical models. Often the partial differential equation links the physical phenomenon to a geometrical model: A soapfilm at rest is modelled by the nonlinear elliptic minimal surface equation, representing a hypersurface of vanishing extrinsic mean curvature in the surrounding space. The vacuum in General Relativity is modelled by a Lorentzian manifold of vanishing intrinsic average curvature as described by the Einstein field equations for the metric. The Einstein equations can be interpreted as a hyperbolic evolution system for the induced metric and extrinsic curvature of a 3-dimesional spacelike hypersurface creating the 4-dimensional spacetime through time.

Interior curvature estimates for hypersurfaces of prescribed mean-curvature

We prove that a smooth solution of the prescribed mean curvature equation div (v−1 Du) = H, v = (1 + |Du|2)1/2 satisfies an interior curvature estimate of the form |A|v(0)≦cR−1supvBR(0).

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.

Convex ancient solutions of the mean curvature flow

We study solutions of the mean curvature flow which are defined for all negative 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.

Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.

Mini-Workshop: Aspects of Ricci-Flow

The workshop studies Hamilton-Ricciflow of Riemannian metrics on 3-manifolds. The participants give detailed technical lectures on recent work of G. Perelman concerning a priori estimates and surgeries during the flow. The workshop was able to verify major sections of Perelmans work and identified points that need a more detailed exposition.