The moduli space of two-convex embedded tori

Abstract

In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset {\mathbb{R}}^{n+1}$. We prove that for $n\ge 3$ this moduli space is path-connected, and that for $n=2$ the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity.

12 pages

References in corpus (2)

• Mean curvature flow with surgery
• The moduli space of two-convex embedded spheres

Cited by in corpus (3)

• Low Entropy and the Mean Curvature Flow with Surgery
• Mass Drop and Multiplicity in Mean Curvature Flow
• Mean curvature flow in homology and foliations of hyperbolic $3$-manifolds