Helicoidal surfaces rotating/translating under the mean curvature flow

Geometriae Dedicata (Impact Factor: 0.47). 06/2011; DOI: 10.1007/s10711-012-9716-2
Source: arXiv

ABSTRACT We describe all possible self-similar motions of immersed hypersurfaces in
Euclidean space under the mean curvature flow and derive the corresponding
hypersurface equations. Then we present a new two-parameter family of immersed
helicoidal surfaces that rotate/translate with constant velocity under the
flow. We look at their limiting behaviour as the pitch of the helicoidal motion
goes to 0 and compare it with the limiting behaviour of the classical
helicoidal minimal surfaces. Finally, we give a classification of the immersed
cylinders in the family of constant mean curvature helicoidal surfaces.

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    ABSTRACT: As first observed in a 1989 paper by Korevaar, Kusner, and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields. In Theorem 3.5 here, we generalize that law by relaxing the topological restrictions assumed in [KKS], and by allowing a weighted mean curvature functional. We also prove a partial converse (Theorem 4.1). Roughly, it states that when flux is conserved along enough Killing fields, a hypersurface splits into two regions: one with constant (weighted) mean curvature, and one fixed by the given Killing fields. We demonstrate the use of our theory by using it to derive a first integral for twizzlers, i.e, helicoidal surfaces of constant mean curvature in euclidean 3-space.
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    ABSTRACT: In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.
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    ABSTRACT: We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems for translating solitons in the Euclidean space in higher codimension.


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