Closed self-shrinking surfaces in ${\mathbb{R}}^3$ via the torus

Abstract

We construct many closed, embedded mean curvature self-shrinking surfaces ${\Sigma }_g^2\subseteq {\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$: 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}.

30 pages; fixed typos, added 4 figures (5 figures in total)

References in corpus (2)

• Doubling and Desingularization Constructions for Minimal Surfaces
• Smooth compactness of self-shrinkers

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