Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Abstract

Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type Ⅱ singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in \begin{document}$\mathbb{R}^{3}$\end{document} and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [ 10 ]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in \begin{document}$\Bbb R^{n+1}$\end{document} and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.