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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

For a mean curvature flow of complete graphical hypersurfaces M_t=graph u(⋅,t) defined over domains Ω_t, the enveloping cylinder is ∂Ω_t×R. We prove the smooth convergence of M_th e_{n+1} to the enveloping cylinder under certain circumstances. Moreover, we give examples demonstrating that there is no uniform curvature bound in terms of the inital curvature and the geometry of Ω_t. Furthermore, we provide an example where the hypersurface increasingly oscillates towards infinity in both space and time. It has unbounded curvature at all times and is not smoothly asymptotic to the enveloping cylinder. We also prove a relation between the initial spatial asymptotics at the boundary and the temporal asymptotics of how the surface vanishes to infinity for certain rates in the case Ω_t are balls.

Related project(s):
29Curvature flows without singularities

We consider the evolution of hypersurfaces in R^{n+1} with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the H-flow is established by means of approximation by bounded problems.

Related project(s):
29Curvature flows without singularities

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on Sáez' and Schnürer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.

Related project(s):
29Curvature flows without singularities

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→∞.

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Team Members

Dr. Friederike Dittberner
Researcher
Universität Konstanz

Wolfgang Maurer
Doctoral student
Universität Konstanz