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

Abstract For smooth mappings of the unit disc into oriented Grassmannian manifold $${\mathbb {G}}_{n,2}$$ G n , 2 , Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured global existence Coulomb frames with bounded conformal factor provided integral $$|{{\varvec{A}}} |^2$$ | A squared-length second fundamental form, is less than $$\gamma _n=8\pi $$ γ = 8 π . It has since been shown that optimal bounds guarantee this result are: _3 = 8\pi 3 _n 4\pi 4 for $$n \ge 4$$ ≥ isothermal immersions in {R}}^3$$ R hypothesis equivalent to saying sum squares principal curvatures _3$$ The goal here prove when $$n=3$$ same conclusion holds under weaker hypotheses. In particular, it integrable $$|K |$$ K where K Gauss curvature, $$4\pi Since $$2|K |\le |{{\varvec{A}}} ≤ implies known immersions, but may be small large. method, which purely analytic, then developed examine case $$|\varvec{A} only square-integrable. possibility extending language manifolds $$n>3$$ > outlined an Appendix.