Abstract
We prove the existence of metrics with prescribed Q-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on \({\mathbb {R}}^n\) can be realized as the Q-curvature of some Riemannian metric.
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Baird, P., Fardoun, A., Regbaoui, R.: Prescribed \(Q\)-curvature on manifolds of even dimension. J. Geom. Phys. 59(2), 221–233 (2009)
Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969)
Branson, T.P.: Differential operator scanonically associated to a conformal structure. Math. Scand. 57, 293–345 (1985)
Branson, T., Gilkey, P., Pohjanpelto, J.: Invariants of locally conformally flat manifolds. Trans. Am. Math. Soc. 347, 939–953 (1995)
Brendle, S.: Global existence and convergence for a higher order flow in conformal geometry. Ann. Math. (2) 158, 323–343 (2003)
Brendle, S.: Convergence of the \(Q\)-curvature flow on \({\mathbb{S}}^4\). Adv. Math. 205, 1–32 (2006)
Canzani, Y., Gover, R., Jakobson, D., Ponge, R.: Conformal invariants from nodal sets. I. Negative eigen values and curvature prescription. Int. Math. Res. Notice IMRN 9, 2356–2400 (2014)
Case, J., Lin, Y., Yuan, W.: Conformally variational riemannian invariants. Trans. Am. Math. Soc. 371(11), 8217–8254 (2019)
Chang, S.Y.A., Eastwood, M., Ørsted, B., Yang, P.: What is \(Q\)-curvature? Acta Appl. Math. 102(2–3), 119–125 (2008)
Chang, S.Y.A., Yang, P.C.: Extremal metrics of zeta function determinants on 4-manifolds. Ann. Math. 142, 171–212 (1995)
Chang, S.-Y.A., Gursky, M., Yang, P.: Remarks on a fourth order invariant in conformal geometry. Aspects Math. HKU. 353–372 (2019)
Chtioui, H., Rigane, A.: On the prescribed \(Q\)-curvature problem on \({\mathbb{S}}^n\). J. Funct. Anal. 261, 2999–3043 (2011)
Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant \(Q\)-curvature. Ann. Math. 168(3), 813–858 (2008)
Delanoë, P., Robert, F.: On the local Nirenberg problem for the \(Q\)-curvatures. Pacific J. Math. 231, 293–304 (2007)
Fefferman, C., Graham, C.R.: \(Q\)-curvature and Poincaré metrics. Math. Res. Lett. 9, 139–151 (2002)
Fischer, A., Marsden, J.: Linearization stability of nonlinear partial differential equations. In: Proceedings of a Symposium in Pure Mathematics, vol. 27. American Mathematical Society, Providence, pp. 219–263 (1975)
Fisher, A.: Mardsen, Linearization stability of nonlinear partial differential equations. In: Proceedings of a Symposium in Pure Mathematics, vol. 27, Part 2. pp. 219–263 (1975)
Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)
Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian I Existence. J. Lond. Math. Soc. (2) 46(3), 557–565 (1992)
Gursky, M.: The principal eigenvalue of a conormally invariant differential operator, with an application to semilinear elliptic PDE. Commun. Math. Phys. 207, 131–147 (1999)
Kazdan, J.: Prescribing the Curvature of a Riemannian Manifold. American Mathematics Society, New York (1984) (CBMS Regional Conference Series 57)
Kazdan, J., Warner, F.: Curvature functions for compact 2-manifold. Ann. Math. 99, 14–47 (1974)
Kazdan, J., Warner, F.: Curvature functions for open 2-manifold. Ann. Math. 99, 203–219 (1974)
Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. (2) 101, 317–331 (1975)
Kazdan, J., Warner, F.: Scalar Curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)
Kazdan, J., Warner, F.: A direct approach to the determination of Gaussian and scalar curvature functions. Invent. Math. 28, 227–230 (1975)
Lin, J., Yuan, W.: A symmetric 2-tensor canonically associated to Q-curvature and its applications. Pacific J. Math. 291, 425–438 (2017)
Levy, T., Oz, Y.: Liouville conformal field theories in higher dimensions (2018). arXiv:1804.02283. [hep-th]
Lin, Y.-J., Yuan, W.: Deformations of \(Q\)-curvature I. Calc. Var. Partial Differ. Equ. 55(4):Paper No. 101, 29 (2016)
Malchiodi, A., Struwe, M.: \(Q\)-curvature flow on \({\mathbb{S}}^4\). J. Differ. Geom. 73, 1–44 (2006)
Mazýa, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Monographs and Studies in Mathematics, vol. 23. Pitman, Boston (1985)
Nakayama, Y.: Canceling the Weyl anomaly from a position-dependent coupling. Phys. Rev. D 97(4), 045008 (2018). https://doi.org/10.1103/PhysRevD.97.045008. arXiv:1711.06413
Ndiaye, C.B.: Constant \(Q\)-curvature metrics in arbitrary dimension. J. Funct. Anal. 251(1), 1–58 (2007)
Robert, F.: Admissible \(Q\)-Curvatures Under Isometries for the Conformal GJMS Operators. Contemporary Mathematics, vol. 540, pp. 241–259. American Mathematics Society, Providence (2011)
Wallach, N., Warner, F.: Curvature forms for 2-manifolds. Proc. Am. Math. Soc. 25, 712–713 (1970)
Wei, J., Xu, X.: On conformal deformations of metrics on \({\mathbb{S}}^n\). J. Funct. Anal. 157, 292–325 (1998)
Acknowledgements
T.C would like to thank C. Arezzo for his kind interest in this work. We thank the referee for valuable comments and suggestions.
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T. Cruz has been partially suported by CNPq/Brazil Grant 311803/2019-9.
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Cruz, F.F., Cruz, T. On the prescribed Q-curvature problem in Riemannian manifolds. manuscripta math. 165, 121–133 (2021). https://doi.org/10.1007/s00229-020-01198-y
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DOI: https://doi.org/10.1007/s00229-020-01198-y