On Closed Finite Gap Curves in Spaceforms I

TL;DR

This paper proves that closed finite gap curves are dense in the space of all closed curves in ${ m I extbf{R}}^3$ and ${ m I extbf{S}}^3$ under the Sobolev $W^{2,2}$-norm, highlighting their approximation properties.

Contribution

It establishes the density of closed finite gap curves in the space of all closed curves in ${ m I extbf{R}}^3$ and ${ m I extbf{S}}^3$, expanding understanding of their geometric and analytical significance.

Findings

Finite gap curves are dense in the space of all closed curves.

Density is with respect to the Sobolev $W^{2,2}$-norm.

Results apply to curves in ${ m I extbf{R}}^3$ and ${ m I extbf{S}}^3$.

Abstract

We show that the spaces of closed finite gap curves in ${\mathbb R}^3$ and ${\mathbb S}^3$ are dense with respect to the Sobolev $W^{2,2}$-norm in the spaces of closed curves in ${\mathbb R}^3$ respectively ${\mathbb S}^3$.