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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Aubin, T.: Problemes isoperimetriques et espaces de Sobolov. J. Differ. Geom. 11, 573–598 (1976)
Aubin, T.: Equations différentielles non linéaires et Problèmes de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)
Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Heidelberg (1998)
Baird, P., Kamissoko, D.: On constructing biharmonic maps and metrics. Ann. Glob. Anal. Geom. 23(1), 65–75 (2003)
Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds. The London Mathematical Society Monographs (N.S.) No. 29, Oxford University Press, Oxford (2003)
Baird, P., Fardoun, A., Ouakkas, S.: Conformal and semi-conformal biharmonic maps. Ann. Glob. Anal. Geom. 34, 403–414 (2008)
Baird, P., Fardoun, A., Ouakkas, S.: Biharmonic maps from biconformal deformations with respect to isoparametric functions. Differ. Geom. Appl. 50, 155–166 (2017)
Balmus, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. Preprint (2007). arXiv:math/0701155
Bliss, G.: An integral inequality. J. Lond. Math. Soc. 5, 44–46 (1930)
Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of \(S^3\). Int. J. Math. 12(8), 867–876 (2001)
Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres. Isr. J. Math. 130, 109–123 (2002)
Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semi-liner equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–289 (1989)
Chang, S.-Y.A.: Conformal Invariants and Partial Differential Equations. Princeton University Press, Princeton (2004)
Chen, B.-Y., Nagano, T.: Harmonic metrics, harmonic tensors, and Gauss maps. J. Math. Soc. Jpn. 36(2), 295–313 (1984)
Chen, W., Wei, J., Yan, S.: Infinitely many solutions for the Shrödinger equations in \(\mathbb{R}^N\) with critical growth. J. Differ. Equ. 252(3), 2425–2447 (2012)
Gu, C.H.: Conformally flat spaces and solutions to Yang–Mills equations. Phys. Rev. D (3) 21(4), 970–971 (1980)
Habermann, L.: Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures. Lecture Notes in Mathematics, vol. 1743. Springer, Berlin (2000)
Hebey, E.: Compactness and Stability for Nonlinear Elliptic Equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2014)
Jiang, G.Y.: \(2\)-harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986)
Jiang, G.Y.: Some non-existence theorems of \(2\)-harmonic isometric immersions into Euclidean spaces. Chin. Ann. Math. Ser. 8A, 376–383 (1987)
LeBrun, C.: Four-dimensional Einstein mandifolds, and beyond. In: LeBrun, C., Wang, M. (eds.) Surveys in Differential Geometry, vol. VI: Essays on Einstein Manifolds, pp. 247–285. International Press of Boston, Boston (1999)
Lohkamp, J.: Metrics of negative Ricci curvature. Ann. Math. (2) 140, 655–683 (1994)
Loubeau, E., Ou, Y.-L.: Biharmonic maps and morphisms from conformal mappings. Tohoku Math J. 62(1), 55–73 (2010)
Montaldo, S., Oniciuc, C., Ratto, A.: Rotationally symmetric biharmonic maps between models. J. Math. Anal. Appl. 431, 494–508 (2015)
Ou, Y.-L.: On conformal biharmonic immersions. Anal. Glob. Anal. Geom. 36, 133–142 (2009)
Ou, Y.-L.: Biharmonic conformal immersions into 3-dimensional manifolds. Mediterr. J. Math. 12(2), 541–554 (2015)
Ouakkas, S.: Géométrie conforme associé à quelques opérateurs d’ordre 4. Thesis Université de Brest (2008)
Ouakkas, S.: Biharmonic maps, conformal deformations and the Hopf maps. Differ. Geom. Appl. 26(5), 495–502 (2008)
Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
Vétois, J., Wang, S.: Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four. Advances in Nonlinear Analysis (2017), ISSN (Online) 2191-950X, ISSN (Print) 2191-9496. https://doi.org/10.1515/anona-2017-0085
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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, YL. 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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DOI: https://doi.org/10.1007/s12220-018-0004-8