A Universal Bound for Lower Neumann Eigenvalues of the Laplacian


Let M be an n-dimensional (n ⩾ 2) simply connected Hadamard manifold. If the radial Ricci curvature of M is bounded from below by (n − 1)k(t) with respect to some point pM, where t = d(·, p) is the Riemannian distance on M to p, k(t) is a nonpositive continuous function on (0, ∞), then the first n nonzero Neumann eigenvalues of the Laplacian on the geodesic ball B(p, l), with center p and radius 0 < l < ∞, satisfy

$${1 \over {{\mu _1}}} + {1 \over {{\mu _2}}} + \ldots + {1 \over {{\mu _n}}}\geqslant {{{l^{n + 2}}} \over {(n + 2)\int_0^l {{f^{n - 1}}(t){\rm{d}}t} }},$$

where f(t) is the solution to

$$\left\{ {\matrix{ {f(t) + k(t)f(t) = 0\;\;\;\;{\rm{on}}\;(0,\infty ),} \hfill \cr {f(0) + 0,\;\;f\prime (0) = 1.} \hfill \cr } } \right.$$

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


  1. [1]

    M. S. Ashbaugh, R. D. Benguria: Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions. SIAM J. Math. Anal. 24 (1993), 557–570.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. S. Ashbaugh, R. D. Benguria, R. S. Laugesen, T. Weidl: Low eigenvalues of Laplace and Schrödinger operators. Oberwolfach Rep. 6 (2009), 355–428.

    MathSciNet  Article  Google Scholar 

  3. [3]

    C. Bandle: Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane. SIAM J. Appl. Math. 22 (1972), 142–147.

    MathSciNet  Article  Google Scholar 

  4. [4]

    C. Bandle: Isoperimetric Inequalities and Applications. Monographs and Studies in Mathematics 7, Pitman, Boston, 1980.

    Google Scholar 

  5. [5]

    L. E. J. Brouwer: Über Abbildung von Mannigfaltigkeiten. Math. Ann. 71 (1911), 97–115. (In German.)

    MathSciNet  Article  Google Scholar 

  6. [6]

    I. Chavel: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics 115, Academic Press, Orlando, 1984.

    Google Scholar 

  7. [7]

    C. Enache, G. A. Philippin: Some inequalities involving eigenvalues of the Neumann Laplacian. Math. Methods Appl. Sci. 36 (2013), 2145–2153.

    MathSciNet  Article  Google Scholar 

  8. [8]

    P. Freitas, J. Mao, I. Salavessa: Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds. Calc. Var. Partial Differ. Equ. 51 (2014), 701–724.

    MathSciNet  Article  Google Scholar 

  9. [9]

    A. Girouard, N. Nadirashvili, I. Polterovich: Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83 (2009), 637–662.

    MathSciNet  Article  Google Scholar 

  10. [10]

    J. Mao: Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel. J. Math. Pures Appl. 101 (2014), 372–393.

    MathSciNet  Article  Google Scholar 

  11. [11]

    E. H. Spanier: Algebraic Topology. McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1966.

    Google Scholar 

  12. [12]

    G. Szegö: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3 (1954), 343–356.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    H. F. Weinberger: An isoperimetric inequality for the n-dimensional free membrane problem. J. Ration. Mech. Anal. 5 (1956), 633–636.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    C. Xia: A universal bound for the low eigenvalues of Neumann Laplacians on compact domains in a Hadamard manifold. Monatsh. Math. 128 (1999), 165–171.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Jing Mao.

Additional information

This research was supported in part by the NSF of China (Grant No. 11401131), China Scholarship Council, the Fok Ying-Tung Education Foundation (China), and Hubei Key Laboratory of Applied Mathematics (Hubei University).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lu, W., Mao, J. & Wu, C. A Universal Bound for Lower Neumann Eigenvalues of the Laplacian. Czech Math J 70, 473–482 (2020). https://doi.org/10.21136/CMJ.2019.0363-18

Download citation


  • Hadamard manifold
  • Neumann eigenvalue
  • radial Ricci curvature

MSC 2010

  • 35P15
  • 53C20