29

Curvature flows without singularities

Uniform estimates. The evolution of a family of (graphical) hypersurfaces $$(M_t)_{0\le t\le T}$$ in $${\mathbb R}^{n+2}$$ with normal velocity $$F$$ depending on the principal curvatures $$(\lambda_i)_{1\le i\le n+1}$$ of the evolving hypersurfaces is described by the equations

$$\dot{X}=-F\nu$$   or   $$\dot{u}=\sqrt{1+\mid Du\mid^2}\cdot F$$.

The corresponding initial value problem with locally Lipschitz initial data $$u(.,0)=u_0\colon{\mathbb R}^{n+1}\to{\mathbb R}$$was solved by Ecker and Huisken for the mean curvature flow, i.e. for $$F=H=\lambda_1+\ldots+\lambda_{n+1}$$. They prove smooth longtime existence and obtain convergence to homothetically expanding solutions for certain initial data. The project aims to show that there exists a mean curvature flow without singularities $$(M_t)_{t\ge 0}$$, such that all the principal curvatures of $$M_t$$ are bounded in terms of the geometry (principal curvatures and size of a tubular neighborhood) of $$\partial\Omega_t$$. Such estimates shall be applied to establish the existence of a mean curvature flow without singularities subject to von Neumann boundary conditions for $$\partial\Omega_t$$ on a supporting hypersurface.

Fully nonlinear flows. Graphical solutions defined on subsets evolving by some elementary symmetric function of the principal curvatures yield a corresponding evolution of the domains of definition $$\Omega_t$$. The project aims to show that there exists a curvature flow without singularities for fully nonlinear curvature flows. We want to impose some natural conditions on the normal velocities $$F$$, but will allow $$F$$  to be equal to (positive) powers of $$S_k$$.

Translating solutions. Translating solutions to mean curvature flow arise as limits of type II singularities in mean curvature flow. The project aims to find new graphical examples of translating solutions to mean curvature flow on time independent sets $$\Omega=\Omega_t$$ with $$H(\partial\Omega)=0$$.

Publications

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F0⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

• 1

Team Members

Dr. Friederike Dittberner
Researcher
Universität Konstanz

Wolfgang Maurer
Doctoral student
Universität Konstanz
wolfgang.maurer(at)uni-konstanz.de