Advertisement

Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 751–795 | Cite as

On the \(L^{r}\) Hodge theory in complete non compact Riemannian manifolds

  • Eric AmarEmail author
Article

Abstract

We study solutions for the Hodge laplace equation \(\Delta u=\omega \) on p forms with \(\displaystyle L^{r}\) estimates for \(\displaystyle r>1.\) Our main hypothesis is that \(\Delta \) has a spectral gap in \(\displaystyle L^{2}.\) We use this to get non classical \(\displaystyle L^{r}\) Hodge decomposition theorems. An interesting feature is that to prove these decompositions we never use the boundedness of the Riesz transforms in \(\displaystyle L^{s}.\) These results are based on a generalisation of the Raising Steps Method to complete non compact Riemannian manifolds.

References

  1. 1.
    Amar, E.: The raising steps method. Application to the \(\bar{\partial }\) equation in Stein manifolds. J. Geom. Anal. 26(2), 898–913 (2016). doi: 10.1007/s12220-015-9576-8
  2. 2.
    Amar, E.: The raising steps method. Applications to the \({L}^{r}\) Hodge theory in a compact Riemannian manifold. HAL-01158323 (2015)Google Scholar
  3. 3.
    Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, Berlin (1971)Google Scholar
  4. 4.
    Bergh, J., LöfStröm, J.: Interpolation Spaces, vol.223. Grundlehren der mathematischen Wissenchaften (1976)Google Scholar
  5. 5.
    Bueler, E.L.: The heat kernel weigted Hodge Laplacian on non compact manifolds. Trans. Am. Math. Soc. 351(2), 683–713 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Donnelly, H.: The diffential form spectrum of hyperbolic spaces. Manuscr. Math. 33, 365–385 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  8. 8.
    Gaffney, M.P.: Hilbert space methods in the theory of harmonic integrals. Am. Math. Soc. 78, 426–444 (1955)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations. Grundlheren der mathematischen Wissenschaften, vol. 224. Springer, Berlin (1998)Google Scholar
  10. 10.
    Gromov, M.: Kähler hyperbolicity and \({L^2}\)-Hodge theory. J. Differ. Geom. 33, 253–320 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Guneysu, B., Pigola, S.: Calderon-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds. Adv. Math. 281, 353–393 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hebey, E.: Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics, vol. 1635. Springer, Berlin (1996)Google Scholar
  13. 13.
    Hebey, E., Herzlich, M.: Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds. Rend. Mat. Appl. (7) 17(4), 569–605 (1997)Google Scholar
  14. 14.
    Kodaira, K.: Harmonic fields in Riemannian manifolds (generalized potential theory). Ann. Math. 50, 587–665 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Li, X.-D.: On the strong \({L}^p\)-Hodge decomposition over complete riemannian manifolds. J. Funct. Anal. 257, 3617–3646 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Li, X.D.: \({L}_p\)-estimates and existence theorems for the \(\bar{\partial }\)-operator on complete kähler manifolds. Adv. Math. 224, 620–647 (2010)Google Scholar
  17. 17.
    Li, X.D.: Riesz transforms on forms and \({L}_p\)-hodge decomposition on complete riemannian manifolds. Rev. Mat. Iberoam. 26(2), 481–528 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Li, X.D.: Sobolev inequalities on forms and \({L}_{p, q}\)-cohomology on complete riemannian manifolds. J. Geom. Anal. 20, 354–387 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lohoué, N.: L’équation de Poisson pour les formes différentielles sur un espace symétrique et ses applications. Bull. Sci. Math. 140, 11–57 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lott, J.: The zero-in-the-spectrum question. Enseign. Math. 42, 341–376 (1996)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Scott, C.: \({L}^p\) theory of differential forms on manifolds. Trans. Am. Math. Soc. 347(6), 2075–2096 (1995)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Taylor, M.E. Differential Geometry. Course of the University of North Carolina. www.unc.edu/math/Faculty/met/diffg.html
  23. 23.
    Varopoulos, N.: Small time gaussian estimates of heat diffusion kernels. Bull. Sci. Math. 113, 253–277 (1989)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Université de BordeauxTalenceFrance

Personalised recommendations