Biharmonic Conformal Maps in Dimension Four and Equations of Yamabe-Type

Abstract

We prove that the problem of constructing biharmonic conformal maps on a 4-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition, we characterize all solutions on Euclidean 4-space and show that there exists at least one proper biharmonic conformal map from any closed Einstein 4-manifold of negative Ricci curvature.

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Correspondence to Paul Baird.

Additional information

Ye-Lin Ou was supported by grant \(\#427231\) from the Simons Foundation. The author is also grateful to the Université de Bretagne Occidentale and the Laboratoire de Mathématiques de Bretagne Atlantique for their hospitality during a visit in May 2017 during which time most of this work was done. The authors express their thanks to Jérome Vétois and to Emmanuel Hebey for providing answers to questions related to Yamabe-type equations with large potentials. They also express their thanks to the referees whose suggestions have helped to improve the presentation of this paper.

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Baird, P., Ou, Y. Biharmonic Conformal Maps in Dimension Four and Equations of Yamabe-Type. J Geom Anal 28, 3892–3905 (2018). https://doi.org/10.1007/s12220-018-0004-8

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Keywords

  • Biharmonic map
  • Conformal biharmonic map
  • Einstein 4-manifold
  • Yamabe equation
  • Möbius transformation

Mathematics Subject Classification

  • 58E20
  • 53A30