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

Pavel I. Plotnikov, John F. Toland

Research output: Contribution to journalArticlepeer-review


For smooth mappings of the unit disc into the oriented Grassmannian manifold Gn,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 γn= 8 π. It has since been shown that the optimal bounds that guarantee this result are: γ3= 8 π and γn= 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.

Original languageEnglish
Pages (from-to)1803-1833
Number of pages31
JournalAnnali di Matematica Pura ed Applicata
Issue number4
Early online date30 Jan 2023
Publication statusPublished - 31 Aug 2023


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

ASJC Scopus subject areas

  • Applied Mathematics


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