Quasiconformal flows on non-conformally flat spheres

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We study integral curvature conditions for a Riemannian metric g on S4 that quantify the best bilipschitz constant between (S4,g) and the standard metric on S4. Our results show that the best bilipschitz constant is controlled by the L2-norm of the Weyl tensor and the L1-norm of the Q-curvature, under the conditions that those quantities are sufficiently small, g has a positive Yamabe constant and the Q-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on S4.

Original languageEnglish (US)
Article number108373
JournalAdvances in Mathematics
StatePublished - Aug 6 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Conformal geometry
  • Q-curvature
  • Quasiconformal maps


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