Existence and geometric structure of metrics on surfaces which extremize eigenvalues

Article

Abstract

This is an exposition of the research area around our lecture at the 60th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in [10].

Keywords

spectral theory minimal surface free boundary condition Steklov eigenvalue 

Mathematical subject classification

53A10 58C40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Besson. Sur la multiplicité de la première valeur propre des surfaces riemanniennes. Ann. Inst. Fourier (Grenoble), 30(1) (1980), x, 109–128.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    R. Brooks and E. Makover. Riemann surfaces with large first eigenvalue. J. Anal. Math., 83 (2001), 243–258.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    P. Buser, M. Burger and J. Dodziuk. Riemann surfaces of large genus and large λ 1. Geometry and analysis on manifolds (Katata/Kyoto, 1987), 54–63, Lecture Notes in Math., 1339, Springer, Berlin (1988).CrossRefGoogle Scholar
  4. [4]
    S.Y. Cheng. Eigenfunctions and nodal sets. Comment. Math. Helv., 51(1) (1976), 43–55.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    F. Da Lio and T. Riviére. Three-term commutator estimates and the regularity of \(\tfrac{1} {2}\) -harmonic maps into spheres. Anal. PDE, 4 (2011), 149–190.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    A. El Soufi, H. Giacomini and M. Jazar. A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle. Duke Math. J., 135 (2006), 181–202.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    A. El Soufi and S. Ilias. Immersionsminimales, première valeur propre du laplacien et volume conforme. Math. Ann., 275(2) (1986), 257–267.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    A. El Soufi and S. Ilias. Riemannian manifolds admitting isometric immersions by their first eigenfunctions. Pacific J. Math., 195(1) (2000), 91–99.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math., 226(5) (2011), 4011–4030.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces in the ball, arXiv:1209.3789 [math.DG] (2012).Google Scholar
  11. [11]
    A. Girouard. Fundamental tone, concentration of density, and conformal degeneration on surfaces. Canad. J. Math., 61 (2009), 548–565.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    A. Girouard and I. Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci., 19 (2012), 77–85.MATHMathSciNetGoogle Scholar
  13. [13]
    J. Hersch. Quatre propriétés isopérimétriqes de membranes sphériques homogènes. C.R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1645–A1648.MathSciNetGoogle Scholar
  14. [14]
    D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigram and I. Polterovich. How large can the first eigenvalue be on a surface of genus two? IMRN, 63 (2005), 3967–3985.CrossRefGoogle Scholar
  15. [15]
    D. Jakobson, N. Nadirashvili and I. Polterovich. Extremal metric for the first eigenvalue on a Klein bottle. Cand. J. Math., 58 (2006), 381–400.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    P. Jammes. Prescription du spectre de Steklov dans une classe conforme, arXiv:1209.4571 [math.DG] (2012).Google Scholar
  17. [17]
    M. Karpukhin, G. Kokarev and I. Polterovich. Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces, arXiv:1209.4869v2 [math.DG] (2013).Google Scholar
  18. [18]
    G. Kokarev. Variational aspects of Laplace eigenvalues on Riemannian surfaces, arXiv:1103.2448 [math.SP] (2011).Google Scholar
  19. [19]
    G. Kokarev and N. Nadirashvili. On first Neumann eigenvalue bounds for conformal metrics. Around the research of Vladimir Maz’ya. II, 229–238, Int. Math. Ser. (N. Y.), 12, Springer, New York (2010).CrossRefGoogle Scholar
  20. [20]
    N. Korevaar. Upper bounds for eigenvalues of conformal surfaces. J. Diff. Geom., 37 (1993), 73–93.MATHMathSciNetGoogle Scholar
  21. [21]
    P. Li and S.-T. Yau. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math., 69(2) (1982), 269–291.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    S. Montiel and A. Ros. Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math., 83(1) (1985), 153–166.CrossRefMathSciNetGoogle Scholar
  23. [23]
    N. Nadirashvili. Multiple eigenvalues of the Laplace operator, (Russian) Mat. Sb. (N.S.), 133(175) (1987), 223–237; translation in Math. USSR-Sb., 61 (1988), 225–238.Google Scholar
  24. [24]
    N. Nadirashvili. Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal., 6(5) (1996), 877–897.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    N. Nadirashvili and Y. Sire. Conformal spectrum and harmonic maps, arXiv:1007.3104v3 [math.DG], (2011).Google Scholar
  26. [26]
    J.C.C. Nitsche. Stationary partitioning of convex bodies. Arch. Rational Mech. Anal., 89(1) (1985), 1–19.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    M-F. Vignéras. Quelques remarques sur la conjecture \(\lambda _1 \geqslant \tfrac{1} {4}\). Seminar on Number Theory, Paris 1981–82, Progr. Math., vol. 38, Birkhauser, Boston, MA, (1983), 321–343.Google Scholar
  28. [28]
    R. Weinstock. Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745–753.MATHMathSciNetGoogle Scholar
  29. [29]
    P. Yang and S.-T. Yau. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7(1) (1980), 55–63.MATHMathSciNetGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations