# On closed finite gap curves in spaceforms II

## TL;DR

This paper demonstrates that closed finite gap curves are densely populated within the space of all closed Sobolev curves in hyperbolic 3-space and 2-dimensional space forms, highlighting their prevalence.

## Contribution

It establishes the $W^{2,2}$-density of closed finite gap curves in the Sobolev space of all closed curves in hyperbolic 3-space and 2D space forms, extending understanding of their distribution.

## Key findings

- Finite gap curves are $W^{2,2}$-dense in hyperbolic 3-space.
- Finite gap curves are $W^{2,2}$-dense in 2D space forms.
- Results extend the understanding of the structure of closed curves in spaceforms.

## Abstract

We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^3$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^3$. We also show that the set of closed finite gap curves in any 2-dimensional space form $\mathbb{E}^2$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{E}^2$.