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_n,2, 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| ², the squared-length of the second fundamental form, is less than γ_n= 8 π. It has since been shown that the optimal bounds that guarantee this result are: γ₃= 8 π and γ_n= 4 π for n≥ 4. For isothermal immersions in R³, this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than γ₃. 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| ² is integrable and the integral of | K| , where K is the Gauss curvature, is less than 4 π. Since 2 | K| ≤ | A| ² this implies the known result for isothermal immersions, but | K| may be small when | A| ² 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.