## Abstract

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: γ_{3}= 8 π and γ_{n}= 4 π for n≥ 4. For isothermal immersions in R^{3}, this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than γ_{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 | 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.

Original language | English |
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Pages (from-to) | 1803-1833 |

Number of pages | 31 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 202 |

Issue number | 4 |

Early online date | 30 Jan 2023 |

DOIs | |

Publication status | Published - 31 Aug 2023 |

## Keywords

- Conformal factors
- Geometric measure theory
- Isothermal immersions
- Moving frames
- Willmore energy

## ASJC Scopus subject areas

- Applied Mathematics