Skip to main content
SpringerLink
Log in

A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Let (M, g) be a compact Riemannian n-dimensional manifold with umbilic boundary It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have \(\partial M\) as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is \(n=6,7,8\), provided that the Weyl tensor is always not vanishing on the boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almaraz, S.: A compactness theorem for scalar-flat metrics on manifolds with boundary. Calc. Var. 41, 341–386 (2011)

    Article  MathSciNet  Google Scholar 

  2. Almaraz, S.: Blow-up phenomena for scalar-flat metrics on manifolds with boundary. J. Differ. Equ. 251(7), 1813–1840 (2011)

    Article  MathSciNet  Google Scholar 

  3. Almaraz, S.: An existence theorem of conformal scalar-flat metrics on manifolds with boundary. Pac. J. Math. 248, 1–22 (2010)

    Article  MathSciNet  Google Scholar 

  4. Almaraz, S., Queiroz, O., Wang, S.: A compactness theorem for scalar-flat metrics on 3-manifolds with boundary. J. Funct. Anal. 277, 2092–2116 (2019)

    Article  MathSciNet  Google Scholar 

  5. Ambrosetti, A., Li, Y.Y., Malchiodi, A.: On the Yamabe problem and the scalar curvature problems under boundary conditions. Math. Ann. 322, 667–699 (2002)

    Article  MathSciNet  Google Scholar 

  6. Aubin, T.: Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)

    MathSciNet  MATH  Google Scholar 

  7. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)

    Book  Google Scholar 

  8. Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)

    Article  MathSciNet  Google Scholar 

  9. Chen, S.S.: Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions,arxiv:0912.1302

  10. Disconzi, M., Khuri, M.: Compactness and non-compactness for the Yamabe problem on manifolds with boundary. J. Reine Angew. Math. 724, 145–201 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Escobar, J.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. 136, 1–50 (1992)

    Article  MathSciNet  Google Scholar 

  12. Escobar, J.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37, 687–698 (1988)

    Article  MathSciNet  Google Scholar 

  13. Felli, V.: Ahmedou, M Ould: Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries. Math. Z. 244, 175–210 (2003)

    Article  MathSciNet  Google Scholar 

  14. Felli, V.: Ahmedou, M Ould: A geometric equation with critical nonlinearity on the boundary. Pac. J. Math. 218, 75–99 (2005)

    Article  Google Scholar 

  15. Ghimenti, M.G., Micheletti, A.M.: Compactness for conformal scalar-flat metrics on umbilic boundary manifolds 200, 30 p. (2020)

  16. Ghimenti, M.G., Micheletti, A.M., Pistoia, A.: Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary. J. Differ. Equ. 267, 587–618 (2019)

    Article  MathSciNet  Google Scholar 

  17. Ghimenti, M., Micheletti, A.M., Pistoia, A.: On Yamabe type problems on Riemannian manifolds with boundary. Pac. J. Math. 284, 79–102 (2016)

    Article  MathSciNet  Google Scholar 

  18. Ghimenti, M., Micheletti, A.M., Pistoia, A.: Linear perturbation of the Yamabe problem on manifolds with boundary. J. Geom. Anal. 28, 1315–1340 (2018)

    Article  MathSciNet  Google Scholar 

  19. Giraud, G.: Sur la problème de Dirichlet généralisé. Ann. Sci. Ècole Norm. Sup. 46, 131–145 (1929)

    Article  MathSciNet  Google Scholar 

  20. Han, Z.C., Li, Y.: The Yamabe problem on manifolds with boundary: existence and compactness results. Duke Math, J. 99, 489–542 (1999)

    Article  MathSciNet  Google Scholar 

  21. Hebey, E., Vaugon, M.: Le probleme de Yamabe equivariant. Bull. Sci. Math. 117, 241–286 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Kim, S., Musso, M., Wei, J.: Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary,arXiv:1906.01317

  23. Khuri, M., Marques, F., Schoen, R.: A compactness theorem for the Yamabe problem. J. Differ. Geom. 81, 143–196 (2009)

    Article  MathSciNet  Google Scholar 

  24. Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Commun. Contemp. Math. 1, 1–50 (1999)

    Article  MathSciNet  Google Scholar 

  25. Mayer, M., Ndiaye, C.B.: Barycenter technique and the Riemann mapping problem of Cherrier-Escobar. J. Differ. Geom. 107(3), 519–560 (2017)

    Article  MathSciNet  Google Scholar 

  26. Marques, F.: Existence results for the Yamabe problem on manifolds with boundary. Indiana Univ. Math. J. 54, 1599–1620 (2005)

    Article  MathSciNet  Google Scholar 

  27. Marques, F.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. J. Differ. Geom. 71, 315–346 (2005)

    Article  MathSciNet  Google Scholar 

  28. Marques, F.: Compactness and non compactness for Yamabe-type problems. Prog. Nonlinear Differ. Equ. Appl. 86, 121–131 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differ. Equ. 4, 1–25 (1996)

    Article  MathSciNet  Google Scholar 

  30. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)

    Article  MathSciNet  Google Scholar 

  31. Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Annali Scuola Norm. Sup. Pisa 22, 265–274 (1968)

    MathSciNet  MATH  Google Scholar 

  32. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56126, Pisa, Italy

    Marco G. Ghimenti & Anna Maria Micheletti

Authors
  1. Marco G. Ghimenti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Maria Micheletti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco G. Ghimenti.

Additional information

Communicated by A. Malchiodi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghimenti, M.G., Micheletti, A.M. A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary. Calc. Var. 60, 119 (2021). https://doi.org/10.1007/s00526-021-01983-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-01983-x

Mathematics Subject Classification

Search

Navigation