Convergence of solutions to the mean curvature flow with a Neumann boundary condition

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Abstract

This work continues our considerations in [15], where we discussed existence and regularity results for the mean curvature flow with homogenious Neumann boundary data. We study the long time evolution of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. On the boundary, a Neumann condition is prescribed in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurfaceΣ. We deduce estimates for the curvature of the immersions and, in a special case, we obtain a precise description of the possible singularities.