Construction of complete embedded self-similar surfaces under mean curvature flow. III
This is the final part of a series of papers, where an original method is developed for constructing new examples of self-shrinkers for the mean curvature flow. See the first part [Trans. Am. Math. Soc. 361, No. 4, 1683--1701 (2009; Zbl 1166.53046)], the second part [Adv. Differ. Equ. 15, No. 5--6, 503--530 (2010; Zbl 1200.53061)], and also [\textit{N. Kapouleas}, J. Differ. Geom. 47, No. 1, 95--169 (1997; Zbl 0936.53006); in: Global theory of minimal surfaces. Proceedings of the Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25--July 27, 2001. Hoffman, David (ed.), Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute. Clay Mathematics Proceedings 2, 489--524 (2005; Zbl 1100.53010)]). The main result is the following: { Theorem.} There exists a natural number \(\tilde m\) so that for any natural number \(m>\tilde m\) there exists a surface \(\tilde M_m\) with the following properties: 1) \(\tilde M_m\) is a complete smooth surface, which satisfies the self-shrinker equation \(\tilde H + \tilde X\cdot\nu\), where \(\tilde X\) is the position vector, \(\tilde H\) is the mean curvature and \(\nu\) is the unit normal vector oriented in such a way that the mean curvature vector is \(\tilde H\nu\). 2) \(\tilde M_m\) is invariant under rotation around the \(\tilde x\)-axis by \(180^\circ\). 3) \(\tilde M_m\) is invariant under reflections across planes containing the \(\tilde z\)-axis and forming angles \(\pi/(2m)+k\pi/m\), \(k\in\mathbb Z\), with the \(\tilde x\)-axis. 4) Let \(U\) be the open top half of the ball of radius 2. As \(m\to\infty\), the sequence of surfaces \(\tilde M_m\) tends to the sphere of radius \(\sqrt{2}\) centered at the origin on any compact set of \(U\). 5) \(\tilde M_m\) is asymptotic to a cone. 6) Let \(T\) stand for the translation by the vector \(-\sqrt{2} \vec e_x\). The sequence of surfaces \[ mT(\tilde M_m) = \left\{m\tilde x, m\tilde y, m\tilde z| (\tilde x+\sqrt{2},\tilde y, \tilde z)\in \tilde M_m\right\} \] converges in \(C^k\) to a Sherk surface on compact sets, for all \(k\in\mathbb N\).
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