Closed self-shrinking surfaces in R^3 via the torus

Details Closed self-shrinking surfaces in R^3 via the torus Closed self-shrinking surfaces in R^3 via the torus

2011-11-30

arxiv.org/abs/1111.7318v1

Mathematics

Differential Geometry

26 pages, 1 figure

Scientific paper

We construct many closed, embedded mean curvature self-shrinking surfaces $\Sigma_g^2\subseteq\Reals^3$ of high genus $g=2k$, $k\in \mathbb{N}$. Each of these shrinking solitons has isometry group equal to the dihedral group on $2g$ elements, and comes from the "gluing", i.e. desingularizing of the singular union, of the two known closed embedded self-shrinkers in $\Reals^3$: The round 2-sphere $\mathbb{S}^2$, and Angenent's self-shrinking 2-torus $\mathbb{T}^2$ of revolution. This uses the results and methods N. Kapouleas developed for minimal surfaces in \cite{Ka97}--\cite{Ka}.

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