Ground state Dirac bubbles and Killing spinors

Abstract

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.

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Notes

  1. 1.

    This is to identify the spinor bundles associated to different spin structures.

References

  1. 1.

    Ammann, B.: A spin-conformal lower bound of the first positive Dirac eigenvalue. Differ. Geom. Appl. 18, 21–32 (2003)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ammann, B.: A variational problem in conformal spin geometry. Universität Hamburg, Habilitationsschift (2003)

    Google Scholar 

  3. 3.

    Ammann, B.: The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions. Commun. Anal. Geom. 17, 429–479 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ammann, B., Grosjean, J.-F., Humbert, E., Morel, B.: A spinorial analogue of Aubin’s inequality. Math. Z. 260, 127–151 (2008)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ammann, B., Humbert, E., Morel, B.: Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds. Commun. Anal. Geom. 14, 163–182 (2006)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Arbunich, J., Sparber, C.: Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures. J. Math. Phys. 59011509, 18 (2018)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bär, C.: Lower eigenvalue estimates for Dirac operators. Math. Ann. 293, 39–46 (1992)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bär, C.: Zero sets of solutions to semilinear elliptic systems of first order. Invent. Math. 138, 183–202 (1999)

    MathSciNet  Article  ADS  Google Scholar 

  9. 9.

    Bartsch, T., Xu, T.: A spinorial analogue of the Brezis–Nirenberg theorem involving the critical Sobolev exponent. ArXiv e-prints (2018)

  10. 10.

    Borrelli, W.: Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity. J. Differ. Equ. 263, 7941–7964 (2017)

    MathSciNet  Article  ADS  Google Scholar 

  11. 11.

    Borrelli, W.: Weakly localized states for nonlinear Dirac equations. Calc. Var. Partial Differ. Equ. 57(57), 155 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Borrelli, W.: Symmetric solutions for a 2D critical Dirac equation. ArXiv e-prints (2020). arXiv:2010.04630

  13. 13.

    Borrelli, W., Frank, R.L.: Sharp decay estimates for critical Dirac equations. Trans. Am. Math. Soc. 373, 2045–2070 (2020)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Borrelli, W., Maalaoui, A.: Some properties of Dirac-Einstein bubbles. J. Geom. Anal. (2020). https://doi.org/10.1007/s12220-020-00503-1

  15. 15.

    Branding, V.: An estimate on the nodal set of eigenspinors on closed surfaces. Math. Z. 288, 1–10 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Caffarelli, L.A., Friedman, A.: The free boundary in the Thomas–Fermi atomic model. J. Differ. Equ. 32, 335–356 (1979)

    MathSciNet  Article  ADS  Google Scholar 

  18. 18.

    Caffarelli, L.A., Friedman, A.: Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations. J. Differ. Equ. 60, 420–433 (1985)

    MathSciNet  Article  ADS  Google Scholar 

  19. 19.

    Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Federer, H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band, vol. 153. Springer, New York (1969)

    Google Scholar 

  21. 21.

    Fefferman, C.L., Weinstein, M.I.: Honeycomb lattice potentials and dirac points. J. Am. Math. Soc. 25, 1169–1220 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Fefferman, C.L., Weinstein, M.I.: Waves in honeycomb structures. Journées équations aux dérivées partielles , 12 (2012). https://doi.org/10.5802/jedp.95

  23. 23.

    Fefferman, C.L., Weinstein, M.I.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326, 251–286 (2014)

    MathSciNet  Article  ADS  Google Scholar 

  24. 24.

    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

    MathSciNet  Article  ADS  Google Scholar 

  25. 25.

    Ginoux, N.: The Dirac spectrum. Lecture Notes in Mathematics, vol. 1976. Springer, Berlin (2009)

  26. 26.

    Grosse, N.: On a conformal invariant of the Dirac operator on noncompact manifolds. Ann. Glob. Anal. Geom. 30, 407–416 (2006)

    MathSciNet  Article  Google Scholar 

  27. 27.

    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)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Isobe, T.: Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds. J. Funct. Anal. 260, 253–307 (2011)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Jost, J.: Riemannian Geometry and Geometric Analysis, Universitext, 6th edn. Springer, Heidelberg (2011)

    Google Scholar 

  31. 31.

    Jost, J., Keßler, E., Tolksdorf, J., Wu, R., Zhu, M.: Regularity of solutions of the nonlinear sigma model with gravitino. Commun. Math. Phys. 358, 1–197 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Jost, J., Keßler, E., Tolksdorf, J., Wu, R., Zhu, M.: Symmetries and conservation laws of a nonlinear sigma model with gravitino. J. Geom. Phys. 128, 185–198 (2018)

    MathSciNet  Article  ADS  Google Scholar 

  33. 33.

    Käenmäki, A., Lehrbäck, J., Vuorinen, M.: Dimensions, Whitney covers, and tubular neighborhoods. Indiana Univ. Math. J. 62, 1861–1889 (2013)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Kim, D.S., Kim, T., Rim, S.-H.: Some identities involving Gegenbauer polynomials. Adv. Differ. Equ. 219, 11 (2012)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Kim, Y.M.: Carleman inequalities for the Dirac operator and strong unique continuation. Proc. Am. Math. Soc. 123, 2103–2112 (1995)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)

    Google Scholar 

  37. 37.

    Lü, H., Pope, C.N., Rahmfeld, J.: A construction of Killing spinors on \(S^n\). J. Math. Phys. 40, 4518–4526 (1999)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Maalaoui, A.: Infinitely many solutions for the spinorial Yamabe problem on the round sphere. Nonlinear Differ. Equ. Appl. 23, 14 (2016)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics. Fractals and rectifiability, vol. 44. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  40. 40.

    Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geom. 6, 247–258 (1971/72)

  41. 41.

    Raulot, S.: A Sobolev-like inequality for the Dirac operator. J. Funct. Anal. 256, 1588–1617 (2009)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Sarri, O.: Spin Geometry, Advanced Topics in Analysis: Sobolev spaces, Online Lecture notes (2019)

  43. 43.

    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971)

    Google Scholar 

  44. 44.

    Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Struwe, M.: Variational methods, vol. 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer, Berlin, fourth ed., 2008. Applications to nonlinear partial differential equations and Hamiltonian systems

  46. 46.

    Swanson, D., Ziemer, W.P.: Sobolev functions whose inner trace at the boundary is zero. Ark. Mat. 37, 373–380 (1999)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Ziemer, W.: Weakly Differentiable Functions. Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)

    Google Scholar 

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Acknowledgements

The authors are grateful to B. Ammann for bringing to their attention some results contained in [3] and to G. Buttazzo for pointing out reference [46] to them. A.M. has been partially supported by the project Geometric problems with loss of compactness from Scuola Normale Superiore and by MIUR Bando PRIN 2015 2015KB9WPT\(_{001}\). A.M. and W.B. are members of GNAMPA as part of INdAM and are supported by the GNAMPA 2020 project Aspetti variazionali di alcune PDE in geometria conforme. W.B. and R.W. are supported by Centro di Ricerca Matematica Ennio de Giorgi.

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Borrelli, W., Malchiodi, A. & Wu, R. Ground state Dirac bubbles and Killing spinors. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-04013-1

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