On the Yang–Yau inequality for the first Laplace eigenvalue

  • Mikhail KarpukhinEmail author


In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus \(\gamma \) and the area. The equality in Yang–Yau’s estimate is attained for \(\gamma =0\) by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for \(\gamma =2\). In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang–Yau’s inequality is strict for all genera \(\gamma >2\). Previously this was only known for \(\gamma =1\). In the second part of the paper we apply Chern-Wolfson’s notion of harmonic sequence to obtain an upper bound on the total branching order of harmonic maps from surfaces to spheres. Applications of these results to extremal metrics for eigenvalues are discussed.



The author is grateful to V. Baranovsky, M. Coppens, A. Neves, I. Polterovich and R. Schoen for fruitful discussions. A special thanks goes to M. Coppens for providing the author with a copy of his thesis [Cop83]. The author thanks V. Medvedev and I. Polterovich for remarks on the preliminary version of the manuscript.


  1. Amm09.
    B. Ammann. The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions. Comm. Anal. Geom., (3)17 (2009), 429–479MathSciNetCrossRefGoogle Scholar
  2. ACGH85.
    E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris. Geometry of algebraic curves. Vol. I. Volume 267 of Grundlehren der Mathemaischen Wissenschaften (Fundamental Principles of Mathematical Sciences), (1985).Google Scholar
  3. ACG11.
    E. Arbarello, M. Cornalba and P.A. Griffiths. Geometry of Algebraic Curves: Volume II with a Contribution by Joseph Daniel Harris., Vol. 268. Springer, Berlin (2011).CrossRefGoogle Scholar
  4. BU83.
    S. Bando and H. Urakawa. Generic properties of the eigenvalue of Laplacian for compact Riemannian manifolds. Tôhoku Math. J., (2)35 (1983), 155–172MathSciNetCrossRefGoogle Scholar
  5. Bar75.
    J. Barbosa. On minimal immersions of \(S^{2}\) into \(S^{2m}\). Trans. Amer. Math. Soc., 210 (1975), 75–106Google Scholar
  6. Ber73.
    M. Berger. Sur les premières valeurs propres des varétés Riemanniennes. Compositio Math., 26 (1973), 129–149MathSciNetzbMATHGoogle Scholar
  7. BJRW88.
    J. Bolton, G.R. Jensen, M. Rigoli and L.W. Woodward. On conformal minimal immersions of \(\mathbb{S}^2\) into \({\mathbb{CP}}^n\). Mathematische Annalen, (4)279 (1988), 599–620Google Scholar
  8. CS03.
    B. Colbois and A. El Soufi. Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ’conformal spectrum’. Ann. Global Anal. Geom., (4)24 (2003), 337–349MathSciNetCrossRefGoogle Scholar
  9. CW85.
    S.S. Chern and J. Wolfson. Harmonic maps of \(\mathbb{S}^2\) into a complex Grassmann manifold. Proceedings of the National Academy of Sciences, (8)82 (1985), 2217–2219Google Scholar
  10. CW87.
    S.S. Chern and J.G. Wolfson. Harmonic maps of the two-sphere into a complex Grassmann manifold II. Annals of Mathematics, (2)125 (1987), 301–335MathSciNetCrossRefGoogle Scholar
  11. CKM.
    D. Cianci, M. Karpukhin and V. Medvedev. On branched minimal immersions of surfaces by first eigenfunctions. To appear in Annals of Global Analysis and Geometry. Published online at Preprint arXiv:1711.05916.
  12. Cop05.
    M. Coppens. Five-gonal curves of genus nine. Collectanea Mathematica, (1)56 (2005), 21–26MathSciNetzbMATHGoogle Scholar
  13. Cop83.
    M. Coppens. One-dimensional linear systems of type II on smooth curves. Ph.D. Thesis, Utrecht, (1983).Google Scholar
  14. SI86.
    A. El Soufi and S. Ilias. Immersions minimales, première valeur propre du Laplacien et volume conforme. Mathematische Annalen, (2)275 (1986), 257–267MathSciNetCrossRefGoogle Scholar
  15. SI08.
    A. El Soufi and S. Ilias. Laplacian eigenvalues functionals and metric deformations on compact manifolds. J. Geom. Phys. (1)58 (2008), 89–104MathSciNetCrossRefGoogle Scholar
  16. SI84.
    A. El Soufi and S. Ilias. Le volume conforme et ses applications d’après Li et Yau, Sém. Théorie Spectrale et Géométrie, Institut Fourier, No. VII. (1984), pp. 1983–1984.Google Scholar
  17. ET88.
    J. Eschenburg and R. Tribuzy. Branch points of conformal mappings of surfaces. Mathematische Annalen, 279 (1988), 621–633MathSciNetCrossRefGoogle Scholar
  18. GH78.
    P. Griffiths and J. Harris. Principles of Algebraic Geometry. Wiley, New York (1978).zbMATHGoogle Scholar
  19. Har13.
    R. Hartshorne. Algebraic geometry, Vol. 52. Springer, Berlin (2013).zbMATHGoogle Scholar
  20. Has11.
    A. Hassannezhad. Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem. Journal of Functional Analysis, (12)261 (2011), 3419–3436MathSciNetCrossRefGoogle Scholar
  21. Her70.
    J. Hersch. Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér A-B 270 (1970), A1645–A1648zbMATHGoogle Scholar
  22. JLNNP05.
    D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam and I. Polterovich. How large can the first eigenvalue be on a surface of genus two? Int. Math. Research Notices, 63 (2005), 3967–3985MathSciNetCrossRefGoogle Scholar
  23. Kar16.
    M. Karpukhin. Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces. Int. Math. Research Notices, 20 (2016), 6200–6209MathSciNetCrossRefGoogle Scholar
  24. Kar.
    M. Karpukhin. Index of minimal spheres and isoperimetric eigenvalue inequalities. Preprint arXiv:1905.03174.
  25. KNPP.
    M. Karpukhin, N. Nadirashvili, A. Penskoi and I. Polterovich. An isoperimetric inequality for Laplace eigenvalues on the sphere. Preprint arXiv:1706.05713.
  26. Kle76.
    S.L. Kleiman. \(r\)-special subschemes and an argument of Severi’s. With an appendix by D. Laksov. Advances in Mathematics, (1)22 (1976), 1–31Google Scholar
  27. Kor93.
    N. Korevaar. Upper bounds for eigenvalues of conformal metrics. J. Differential Geom. (1)37 (1993), 79–93MathSciNetCrossRefGoogle Scholar
  28. Kus89.
    R. Kusner. Comparison surfaces for the Willmore problem. Pacific J. of Math. (2)138 (1989), 317–345MathSciNetCrossRefGoogle Scholar
  29. LY82.
    P. Li and S.-T. Yau. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Inventiones mathematicae, (2)69 (1982), 269–291MathSciNetCrossRefGoogle Scholar
  30. Mat18.
    H. Matthiesen. On some variational problems in geometry, Doctoral dissertation, Universitäts-und Landesbibliothek Bonn, (2018).Google Scholar
  31. MR.
    S. Montiel and A. Ros. Schrödinger operators associated to a holomorphic map. In: Global differential geometry and global analysis. Springer, Berlin, pp. 147–174.Google Scholar
  32. Nad96.
    N. Nadirashvili. Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal, (5)6 (1996), 877–897MathSciNetCrossRefGoogle Scholar
  33. Nad88.
    N.S. Nadirashvili. Multiple eigenvalues of the Laplace operator. Sbornik: Mathematics, (1)61 (1988), 225–238MathSciNetCrossRefGoogle Scholar
  34. NP18.
    N.S. Nadirashvili and A.V. Penskoi. An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane. Geometric and Functional Analysis, (5)28 (2018), 1368–1393MathSciNetCrossRefGoogle Scholar
  35. NS19.
    S. Nayatani and T. Shoda. Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian. Comptes Rendus Mathematique, (1)357 (2019), 84–98MathSciNetCrossRefGoogle Scholar
  36. Oss86.
    R. Osserman. A Survey of minimal surfaces. Dover Publications, Inc., Mineola, New York, (1986).zbMATHGoogle Scholar
  37. Pet14.
    R. Petrides. Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces. Geometric and Functional Analysis, (4)24 (2014), 1336–1376MathSciNetCrossRefGoogle Scholar
  38. Pet18.
    R. Petrides. On the existence of metrics which maximize Laplace eigenvalues on surfaces. Int. Math. Research Notices, 14 (2018), 4261–4355MathSciNetCrossRefGoogle Scholar
  39. Ros06.
    A. Ros. One-sided complete stable minimal surfaces. J. Differential Geom. 74 (2006), 69–92MathSciNetCrossRefGoogle Scholar
  40. YY80.
    P.C. Yang and S.-T. Yau. Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (1)7 (1980), 55–63MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, IrvineIrvineUSA

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