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

Alaa Boukholkhal

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.

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

Abstract

We reformulate the problem of finding conformal immersions of closed Riemannian surfaces in the language of the -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.
Paper Structure (4 sections, 7 theorems, 21 equations)

This paper contains 4 sections, 7 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The $h$-principle: an overview
  3. Conformal immersions relation and Teichmüller space
  4. Proof of the main theorem

Key Result

Theorem 1.2

Let $(\Sigma,g)$ be a closed orientable Riemannian surface, and let $(M,h)$ be a Riemannian manifold of dimension $\geq 3$. The inclusion from the space of smooth conformal immersions of $(\Sigma,g)$ in $M$, to the space of immersions of $\Sigma$ in $M$ induces a bijection on the set of path-connect

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 2.1
  • Definition 2.2: Gromov gromov1986
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Gromov gromov1986
  • Corollary 2.6: Gromov gromov1986
  • Definition 3.2
  • ...and 6 more