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

Abstract

For smooth mappings of the unit disc into the oriented Grassmannian manifold {mathbb {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, the squared-length of the second fundamental form, is less than gamma _n=8pi . It has since been shown that the optimal bounds that guarantee this result are: gamma _3 = 8pi and gamma _n = 4pi for n ge 4. For isothermal immersions in {mathbb {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. 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 |{{varvec{A}}} |^2 is integrable and the integral of |K |, where K is the Gauss curvature, is less than 4pi . Since 2|K |le |{{varvec{A}}} |^2 this implies the known result for isothermal immersions, but |K | may be small when |{{varvec{A}}} |^2 is large. The method, which is purely analytic, is then developed to examine the case n=3 when |varvec{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.