A Proof of H\'elein's Conjecture on Boundedness of Conformal Factors when n=3

TL;DR

This paper proves Hélain's conjecture for the case n=3, showing that under weaker conditions involving the integral of the Gauss curvature, bounded conformal factors exist for certain smooth mappings, extending previous results.

Contribution

It establishes the conjecture for n=3 under weaker hypotheses involving the integral of the Gauss curvature, using a purely analytic method.

Findings

The result holds when the integral of |K| is less than 4π.

The proof extends to cases with square-integrable second fundamental form.

The result is sharp, demonstrated by Enneper's surface and stereographic projections.

Abstract

For smooth mappings of the unit disc into the oriented Grassmannian manifold $\mathbb G_{n,2}$, H\'elein (2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $|\boldsymbol A|^2$, the squared-length of the second fundamental form, is less than $\gamma_n=8\pi$. It has since been shown that the optimal bounds on the integral of $|\boldsymbol A|^2$ that guarantee this result are: $\gamma_3 = 8\pi$ and $\gamma_n = 4\pi$ for $n \geq 4$. For isothermal immersions, this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than $\gamma_n$. The goal here is to prove that when $n=3$ the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when $|\boldsymbol A|$ is square-integrable and the integral of $|K|$, $K$ the Gauss curvature, is less than $4\pi$. Since $2|K| \leq |\boldsymbol A|^2$ this implies the known result for isothermal immersions, but $|K|$ may be small when $|\boldsymbol A|^2$ is large. That the result under the weaker hypothesis is sharp is shown by Enneper's surface and stereographic projections. The method, which is purely analytic, is then extended to investigate the case when the length of the second fundamental form is square-integrable.