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

P. I. Plotnikov, J. F. Toland

Published 2020-10-28Version 1

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 length of the second fundamental form, is less than $\gamma_n=8\pi$. Since then it is known that the optimal values are $\gamma_3 = 8\pi$ and $\gamma_n = 4\pi$ for $n \geq 4$. The goal here is to prove, by purely analytic methods, that when $n=3$ the conclusion of H\'elein's Conjecture holds under weaker hypotheses. In the case of isothermal immersions, the new hypothesis is that $|\boldsymbol A|^2$ is integrable and the integral of $|K|$ ($K$ the Gauss curvature) is less than $4\pi$. The original hypothesis on $|\boldsymbol A|^2$ is equivalent to assuming the integral of the sum of the squares of the principal curvatures is less than $8\pi$. Note that although $2|K| \leq |\boldsymbol A|^2$, $|K|$ 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 is then extended to investigate the case when the square integral of the second fundamental form is merely finite.