Abstract
We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac–Einstein equations on \(\mathbb {R}^3\), which appear in the bubbling analysis of conformal Dirac–Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin–Talenti functions, while the spinorial part is the conformal image of \(-\frac{1}{2}\)-Killing spinors on the round sphere \(\mathbb {S}^3\).
Similar content being viewed by others
References
Ammann, B.: The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions. Commun. Anal. Geom. 17, 429–479 (2009)
Ammann, B., Humbert, E., Morel, B.: Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds. Commun. Anal. Geom. 14, 163–182 (2006)
Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, Tech. rep., KANSAS UNIV LAWRENCE, (1956)
Belgun, F.A.: The einstein-dirac equation on sasakian 3-manifolds. J. Geometry Phys. 37, 229–236 (2001)
Borrelli, W.: Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity. J. Differ. Equ. 263, 7941–7964 (2017)
Borrelli, W.: Weakly localized states for nonlinear Dirac equations. Calc. Var. Partial Differ. Equ. 57, 155 (2018)
Borrelli, W., Frank, R.L.: Sharp decay estimates for critical Dirac equations. Trans. Am. Math. Soc. 373, 2045–2070 (2020)
Borrelli, W., Malchiodi, A., Wu, R.: Ground state dirac bubbles and killing spinors, arXiv preprint arXiv:2003.03949, (2020)
Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)
Finster, F., Smoller, J., Yau, S.-T.: Particlelike solutions of the einstein-dirac equations. Phys. Rev. D 59, 104020 (1999)
Ginoux, N.: The Dirac Spectrum. Lecture Notes in Mathematics, vol. 1976. Springer, Berlin (2009)
Greene, R.E., Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27, 265–298 (1974)
Grosse, N.: Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometry. Commun. Partial Differ. Equ. 37, 58–76 (2012)
Guidi, C., Maalaoui, A., Martino, V.: Existence results for the conformal dirac-einstein system, arXiv preprint arXiv:2001.07412, (2020)
Helms, L.L.: Potential Theory, Universitext, 2nd edn. Springer, London (2014)
Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys. 104, 151–162 (1986)
Hijazi, O.: Première valeur propre de l’opérateur de dirac et nombre de yamabe, Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 313, 865–868 (1991)
Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)
Isobe, T.: Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds. J. Funct. Anal. 260, 253–307 (2011)
Jevnikar, A., Malchiodi, A., Wu, R.: Existence results for a super-liouville equation on compact surfaces, arXiv preprint arXiv:1909.12260, (2019)
Jost, J., Wang, G., Zhou, C.: Super-liouville equations on closed riemann surfaces. Commun. Partial Differ. Equ. 32, 1103–1128 (2007)
Jost, J., Wang, G., Zhou, C., Zhu, M.: Energy identities and blow-up analysis for solutions of the super liouville equation. J. Math. et Appl. 92, 295–312 (2009)
Kim, E.C., Friedrich, T.: The einstein-dirac equation on riemannian spin manifolds. J. Geometry Phys. 33, 128–172 (2000)
Kim, Y.M.: Carleman inequalities for the Dirac operator and strong unique continuation. Proc. Am. Math. Soc. 123, 2103–2112 (1995)
Lawson, H.B., Michelsohn, M.-L.: Spin Geometry, Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)
Lü, H., Pope, C.N., Rahmfeld, J.: A construction of Killing spinors on \(S^n\). J. Math. Phys. 40, 4518–4526 (1999)
Maalaoui, A.: Infinitely many solutions for the spinorial Yamabe problem on the round sphere. NoDEA Nonlinear Differ. Equ. Appl. 23, 25 (2016)
Maalaoui, A., Martino, V.: Characterization of the Palais-Smale sequences for the conformal Dirac-Einstein problem and applications. J. Differ. Equ. 266, 2493–2541 (2019)
Maalaoui, A., Martino, V., et al.: Changing-sign solutions for the cr-yamabe equation. Differ. Integral Equ. 25, 601–609 (2012)
Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geometry, 6, 247–258 (1971/72)
Acknowledgements
The authors would like to thank the anonymous referees for their careful reading of the manuscript and their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Borrelli, W., Maalaoui, A. Some Properties of Dirac–Einstein Bubbles. J Geom Anal 31, 5766–5782 (2021). https://doi.org/10.1007/s12220-020-00503-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00503-1