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

Abstract

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