Refinement of Hélein’s conjecture on boundedness of conformal factors when n= 3

For smooth mappings of the unit disc into the oriented Grassmannian manifold G, Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of | A| , the squared-length of the second fundamental form, is less than γ= 8 π. It has since been shown that the optimal bounds that guarantee this result are: γ= 8 π and γ= 4 π for n≥ 4. For isothermal immersions in R, this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than γ. 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 | A| is integrable and the integral of | K| , where K is the Gauss curvature, is less than 4 π. Since 2 | K| ≤ | A| this implies the known result for isothermal immersions, but | K| may be small when | A| is large. The method, which is purely analytic, is then developed to examine the case n= 3 when | A| is only square-integrable. The possibility of extending that result in the language of Grassmannian manifolds to the case n> 3 is outlined in an Appendix.