The $1$-parametric $h$-principle for smooth conformal immersions of surfaces

TL;DR

This paper demonstrates that the space of smooth conformal immersions of closed surfaces is topologically equivalent to the space of all immersions, using the $h$-principle, revealing a fundamental flexibility in such geometric mappings.

Contribution

It reformulates the problem of conformal immersions within the $h$-principle framework and proves a bijection on path components between conformal immersions and all immersions.

Findings

The inclusion map induces a bijection on path connected components.

Conformal immersions are topologically flexible within the space of all immersions.

Abstract

We reformulate the problem of finding conformal immersions of closed Riemannian surfaces in the language of the $h$-principle and we prove that the inclusion from the space of smooth conformal immersions to the space of immersions induces a bijection on the sets of path connected components.