Abstract
In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented n-dimensional (\(n\ge 6\)) Riemannian manifold (M, g) and prove the following results under the condition 
. (1) If (M, g) is locally conformally flat with nonnegative Ricci curvature, then (M, g) is isometric to a quotient of \(\mathbb {R}^n\), \(\mathbb {S}^n\), or \(\mathbb {R}\times \mathbb {S}^{n-1}\). (2) If (M, g) has \(\delta ^2 W=0\) with nonnegative sectional curvature, then (M, g) is isometric to a quotient of the product of Einstein manifolds. Additionally, we investigate some rigidity theorems involving Q-curvature about hypersurfaces in simply-connected space forms. We also show the uniqueness of metrics with constant scalar curvature and constant Q-curvature in a fixed conformal class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bettiol, R., Piccione, P., Sire, Y.: Nonuniqueness of conformal metrics with constant Q-curvature. Int. Math. Res. Not. IMRN 9, 6967–6992 (2021)
Branson, T.: Differential operators canonically associated to a conformal structure. Math. Scand. 57(2), 293–345 (1985)
Carron, G., Herzlich, M.: Conformally flat manifolds with nonnegative Ricci curvature. Compos. Math. 142(3), 798–810 (2006)
Chang, A., Gursky, M., Yang, P.: An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. Math. 155, 709–787 (2002)
Chang, A., Eastwood, M., Ørsted, B., Yang, P.: What is Q-curvature? Acta Appl. Math. 102(2–3), 119–125 (2008)
Cheng, Q.: Compact locally conformally flat Riemannian manifolds. Bull. Lond. Math. Soc. 33(4), 459–465 (2001)
Cheng, S., Yau, S.: Hypersurfaces with constant scalar curvature. Math. Ann. 225(3), 195–204 (1977)
Ceil, T., Ryan, P.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York (2015)
Gursky, M., Hang, F., Lin, Y.: Riemannian manifolds with positive Yamabe invariant and Paneitz operator. Int. Math. Res. Not. IMRN 5, 1348–1367 (2016)
Gursky, M., Malchiodi, A.: A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature. J. Eur. Math. Soc. (JEMS) 17(9), 2137–2173 (2015)
Hang, F., Yang, P.: Q-curvature on a class of 3-manifolds. Commun. Pure Appl. Math. 69(4), 734–744 (2016)
Hang, F., Yang, P.: Q-curvature on a class of manifolds with dimension at least 5. Commun. Pure Appl. Math. 69(8), 1452–1491 (2016)
Hang, F., Yang, P.: Lectures on the fourth-order Q-curvature equation. In: Geometric Analysis Around Scalar Curvatures, vol. 31 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, pp. 1–33. World Scientific Publishing, Hackensack (2016)
Hebey, E., Robert, F.: Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electron. Res. Announc. Am. Math. Soc. 10, 135–141 (2004)
Li, H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305(4), 665–672 (1996)
Li, Y., Xiong, J.: Compactness of conformal metrics with constant Q-curvature. I. Adv. Math. 345, 116–160 (2019)
Lin, Y., Yuan, W.: Deformations of Q-curvature. I. Calc. Var. Partial Differ. Equ. 55(4), Art. 101 (2016)
Lin, Y., Yuan, W.: A symmetric 2-tensor canonically associated to Q-curvature and its applications. Pac. J. Math. 291(2), 425–438 (2017)
Lin, Y., Yuan, W.: Deformations of Q-curvature II. Calc. Var. Partial Differ. Equ. 61(2), Paper No. 74 (2022)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)
Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, vol. 171, 3rd edn. Springer, Cham (2016)
Qing, J., Raske, D.: Compactness for conformal metrics with constant Q-curvature on locally conformally flat manifolds. Calc. Var. Partial Differ. Equ. 26(3), 343–356 (2006)
Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, topics in calculus of variations (Montecatini Terme: Lecture Notes in Mathematics 1365, 1989). Springer, Berlin (1987)
Tian, G., Viaclovsky, J.: Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160(2), 357–415 (2005)
Zhu, S.: The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature. Pac. J. Math. 163(1), 189–199 (1994)
Acknowledgements
We deeply appreciate the referee for providing us with invaluable feedback and insightful suggestions that have significantly improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author has been supported in part by NSFC under Grant Nos. 11501285 and 11871265.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xu, Y., Zhang, S. On some rigidity theorems of Q-curvature. manuscripta math. 174, 535–557 (2024). https://doi.org/10.1007/s00229-023-01506-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-023-01506-2