Abstract
In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface∑. We deduce a very sharp local gradient bound depending only on the curvature of the immersions and∑. Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for anyt>0 or on a maximal finite time interval [0,T] such that the curvature explodes ast→T.
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Stahl, A. Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition. Calc. Var 4, 385–407 (1996). https://doi.org/10.1007/BF01190825
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DOI: https://doi.org/10.1007/BF01190825