Helicoidal surfaces rotating/translating under the mean curvature flow

Geometriae Dedicata (Impact Factor: 0.52). 06/2011; 162(1). DOI: 10.1007/s10711-012-9716-2
Source: arXiv


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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    • "Each of these surfaces is asymptotic at infinity to a vertical translate of the graph of 1 2 r 2 − log(r), where r here denotes the radial distance in R 2 to the origin. (4) Examples of helicoidal type are constructed by Halldorrson in [8]. (5) Examples of infinite genus invariant under a discrete group of translations are constructed by Nguyen in [16] and [17]. "
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    ABSTRACT: We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain what we believe to be the first examples in $\Bbb{R}^3$ of complete embedded translating solitons of the mean curvature flow of arbitrary non-trivial genus. This solves a problem posed by Mart\'in, Savas-Halilaj and Smoczyk.
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    • "Comparing with the theory of minimal surfaces (H = 0) in R 3 , it is natural to impose some geometric property to the surface that makes easier the study of (1), such as that the surface is rotational, helicoidal, ruled or translation. This situation has been studied in [5] [8]. Following the above two motivations, in this paper we consider φ-minimal surfaces of Riemann type and of translation type and that we now explain. "
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    ABSTRACT: We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(\alpha,\beta,\gamma)$ and the surface must be rotational. We also classify all minimal surfaces of translation type.
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    ABSTRACT: Motivated by Ilmanen's correspondence, we present an explicit solution to the prescribed Hoffman-Osserman Gauss map problem for non-minimal translators to the mean curvature flow in Euclidean 4-space. We propose a conjecture on the non-existence of Jenkins-Serrin type unit-speed graphical translators.
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