A Proof of Hélein's Conjecture on Boundedness of Conformal Factors when n=3

Abstract

For smooth mappings of the unit disc into the oriented Grassmannian manifold ${\mathbb{G}}_{n,2}$, Hélein (2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $|\mathbf{A}{|}^2$, 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 $|\mathbf{A}{|}^2$ that guarantee this result are: ${\gamma }_3=8\pi$ and ${\gamma }_n=4\pi$ for $n\ge 4$. 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 $|\mathbf{A}|$ is square-integrable and the integral of $|K|$, $K$ the Gauss curvature, is less than $4\pi$. Since $2|K|\le |\mathbf{A}{|}^2$ this implies the known result for isothermal immersions, but $|K|$ may be small when $|\mathbf{A}{|}^2$ 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.

42 pages