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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 G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)] on the mean curvature flow of convex closed hypersurfaces in \(\mathbb{R}^{n+ 1}\) 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 \(\mathbb{R}^{n+1}\) 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 [A. Stahl, Calc. Var. Partial Differ. Equ. 4, No. 4, 385–407 (1996; Zbl 0851.35053)], 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 \(| A(x, t)|^2\leq K(T_C- t)^{-1}\) 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 \((h_{ij})\geq\varepsilon H(g_{ij})\) 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.

MSC:

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] S.J. Altschuler: Singularities of the curve shrinking flow for space curves. J. Differ. Geom.34 (1991) 491–514 · Zbl 0754.53006
[2] B. Andrews: Contraction of convex hypersurfaces in Euclidian space. Calc. Var.2 (1994) 151–171 · Zbl 0805.35048
[3] S. Angenent: On the formation of singularities in the curve shortening flow. J. Differ. Geom.33 (1991) 601–633 · Zbl 0731.53002
[4] S. Altschuler, L. F. Wu: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var.2 (1994) 101–111 · Zbl 0812.35063
[5] K. Ecker, G. Huisken: Mean curvature evolution of entire graphs. Ann. Math.130 (1989) 453–471 · Zbl 0696.53036
[6] B. Guan: Mean curvature motion of non-parametric hypersurfaces with contact angle condition. Preprint, Indiana University (1994)
[7] R.S. Hamilton: Harmonic Maps of Manifolds with Boundary. Lecture Notes of Mathematics471, Springer 1975 · Zbl 0308.35003
[8] R.S. Hamilton: Convex hypersurfaces with pinched second fundamental form (preprint 1994) · Zbl 0843.53002
[9] R.S. Hamilton: The formation of singularities in the Ricci flow (preprint 1994)
[10] G. Huisken: Flow by mean curvature of convex surfaces into spheres: J. Differ. Geom.20 (1984) 237–266 · Zbl 0556.53001
[11] G. Huisken: Asymptotic behaviour for singularities of the mean curvature flow. J. Differ. Geom.31 (1990) 285–299 · Zbl 0694.53005
[12] G. Huisken: Non-parametric mean curvature evolution with boundary conditions. J. Differ. Eqns.77 (1989) 369–378 · Zbl 0686.34013
[13] A. Stone: A density function and the structure of singularities of the mean curvature flow. Calc. Var.2 (1994) 443–480 · Zbl 0833.35062
[14] A. Stahl: Über den mittleren Krümmungsfluß mit Neumannrandwerten auf glatten Hyperflächen. Dissertation, Universität Tübingen (1994) · Zbl 0869.53002
[15] A. Stahl: Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition (to appear) · Zbl 0851.35053
[16] K. Tso: Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math.38 (1985) 867–882 · Zbl 0612.53005
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