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Convergence of solutions to the mean curvature flow with a Neumann boundary condition. (English) Zbl 0896.35059
The author generalizes well-known results of {\it G. Huisken} [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)] on the mean curvature flow of convex closed hypersurfaces in $\bbfR^{n+ 1}$ to the case of hypersurfaces with boundary, satisfying a Neumann boundary condition on some supporting hypersurface $\Sigma$. A time depending family $F(.,t)$ of embeddings of some $n$-dimensional manifold $M^n$ in $\bbfR^{n+1}$ moves according to the mean curvature flow if ${d\over dt} F(x,t)= \vec H(x,t)$, where $\vec H(.,t)$ is the mean curvature vector of $M_t:= F(M^n, t)$. The Neumann boundary condition along $\Sigma$ requires that $\partial M_t$ is contained in $\Sigma$ and $M_t$ meets $\Sigma$ under a right angle along $\partial M_t$, for all $t$. In a previous work, the author has dealt with the existence and regularity of this flow, while in the present paper he investigates its long time behavior. Complete results are only obtained for the special case that $\Sigma$ is a round sphere or a hyperplane. In this case, it is shown that any strictly convex, compact embedded initial surface $M_0$ which also satisfies the boundary condition shrinks under the flow to a point on $\Sigma$ after finite time $T_C$. Moreover, the second fundamental form $A$ satisfies $| A(x, t)|^2\le K(T_C- t)^{-1}$ for $t\nearrow T_C$ with some constant $K$ and there is a sequence $t_k\to T_C$ such that the appropriately rescaled hypersurfaces $M_{t_k}$ converge in $C^\infty$ to a hemisphere. As the author observes the strong restriction on $\Sigma$ is only needed to conserve the convexity of $M_t$ and the pinching condition $(h_{ij})\ge\varepsilon H(g_{ij})$ with $\varepsilon> 0$ under the flow where $(g_{ij})$ and $(h_{ij})$ are the first and second fundamental forms, respectively. One of the main ingredients of the proof is a weak maximum principle for symmetric 2-tensors satisfying a parabolic inequality which guarantees that such a tensor stays positive along the flow.

35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
35K60Nonlinear initial value problems for linear parabolic equations
58J35Heat and other parabolic equation methods for PDEs on manifolds
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
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