2017, 24: 100-109. doi: 10.3934/era.2017.24.011

Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds

Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9

The author is grateful to Iosif Polterovich for fruitful discussions and comments on the initial versions of the manuscript. The author thanks Spiro Karigiannis for providing the reference [13]

Received  May 04, 2017 Revised  August 16, 2017 Published  September 2017

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.

Citation: Mikhail Karpukhin. Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds. Electronic Research Announcements, 2017, 24: 100-109. doi: 10.3934/era.2017.24.011
References:
[1]

M. Belishev and V. Sharafutdinov, Dirichlet to Neumann operator on differential forms, Bull. Sci. Math., 132 (2008), 128-145.  doi: 10.1016/j.bulsci.2006.11.003.  Google Scholar

[2]

I. Chavel, Riemannian Geometry, A Modern Introduction, 2nd edition, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511616822.  Google Scholar

[3]

A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226 (2011), 4011-4030.  doi: 10.1016/j.aim.2010.11.007.  Google Scholar

[4]

A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Inventiones mathematicae, 203 (2016), 823-890.  doi: 10.1007/s00222-015-0604-x.  Google Scholar

[5]

A. GirouardL. ParnovskiI. Polterovich and D. Sher, The Steklov spectrum of surfaces: Asymptotics and invariants, Math. Proc. Camb. Phil. Soc., 157 (2014), 379-389.  doi: 10.1017/S030500411400036X.  Google Scholar

[6]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. of Spectral Theory, 7 (2017), 321-359.  doi: 10.4171/JST/164.  Google Scholar

[7]

A. Girouard and I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., 19 (2012), 77-85.  doi: 10.3934/era.2012.19.77.  Google Scholar

[8]

A. Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. of Functional Analysis, 261 (2011), 3419-3436.  doi: 10.1016/j.jfa.2011.08.003.  Google Scholar

[9]

J HerschL. Payne and M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.  doi: 10.1007/BF00248412.  Google Scholar

[10]

M. S. Joshi and W. R. B. Lionheart, An inverse boundary value problem for harmonic differential forms, Asymptot. Anal., 41 (2005), 93-106.   Google Scholar

[11]

M. Karpukhin, Steklov problem on differential forms, Preprint, arXiv: 1705.08951. Google Scholar

[12]

G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math., 258 (2014), 191-239.  doi: 10.1016/j.aim.2014.03.006.  Google Scholar

[13]

P. Petersen, Demystifying the Weitzenböck curvature operator Preprint available from: http://www.math.ucla.edu/~petersen/. Google Scholar

[14]

S. Raulot and A. Savo, On the first eigenvalue of the Dirichlet-to-Neumann operator on forms, J. of Functional Analysis, 262 (2012), 889-914.  doi: 10.1016/j.jfa.2011.10.008.  Google Scholar

[15]

G. Schwarz, Hodge Decomposition – A Method for Solving Boundary Value Problems, Lecture Notes in Math., Springer, (1995).  doi: 10.1007/BFb0095978.  Google Scholar

[16]

V. Sharafutdinov and C. Shonkwiler, The complete Dirichlet-to-Neumann map for differential forms, J. Geom. Anal., 23 (2013), 2063-2080.  doi: 10.1007/s12220-012-9320-6.  Google Scholar

[17]

Q. Wang and C. Xia, Sharp bounds for the first non-zero Stekloff eigenvalues, J. of Functional Analysis, 257 (2009), 2635-2644.  doi: 10.1016/j.jfa.2009.06.008.  Google Scholar

[18]

C. Xia, Rigidity for compact manifolds with boundary and non-negative Ricci curvature, Proc. Amer. Math. Soc., 125 (1997), 1801-1806.  doi: 10.1090/S0002-9939-97-04078-1.  Google Scholar

[19]

L. Yang and C. Yu, A higher dimensional generalization of Hersch-Payne-Schiffer inequality for Steklov eigenvalues, J. of Functional Analysis, 272 (2017), 4122-4130.  doi: 10.1016/j.jfa.2017.02.023.  Google Scholar

show all references

References:
[1]

M. Belishev and V. Sharafutdinov, Dirichlet to Neumann operator on differential forms, Bull. Sci. Math., 132 (2008), 128-145.  doi: 10.1016/j.bulsci.2006.11.003.  Google Scholar

[2]

I. Chavel, Riemannian Geometry, A Modern Introduction, 2nd edition, Cambridge University Press, New York, 2006. doi: 10.1017/CBO9780511616822.  Google Scholar

[3]

A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226 (2011), 4011-4030.  doi: 10.1016/j.aim.2010.11.007.  Google Scholar

[4]

A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Inventiones mathematicae, 203 (2016), 823-890.  doi: 10.1007/s00222-015-0604-x.  Google Scholar

[5]

A. GirouardL. ParnovskiI. Polterovich and D. Sher, The Steklov spectrum of surfaces: Asymptotics and invariants, Math. Proc. Camb. Phil. Soc., 157 (2014), 379-389.  doi: 10.1017/S030500411400036X.  Google Scholar

[6]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. of Spectral Theory, 7 (2017), 321-359.  doi: 10.4171/JST/164.  Google Scholar

[7]

A. Girouard and I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., 19 (2012), 77-85.  doi: 10.3934/era.2012.19.77.  Google Scholar

[8]

A. Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. of Functional Analysis, 261 (2011), 3419-3436.  doi: 10.1016/j.jfa.2011.08.003.  Google Scholar

[9]

J HerschL. Payne and M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal., 57 (1975), 99-114.  doi: 10.1007/BF00248412.  Google Scholar

[10]

M. S. Joshi and W. R. B. Lionheart, An inverse boundary value problem for harmonic differential forms, Asymptot. Anal., 41 (2005), 93-106.   Google Scholar

[11]

M. Karpukhin, Steklov problem on differential forms, Preprint, arXiv: 1705.08951. Google Scholar

[12]

G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math., 258 (2014), 191-239.  doi: 10.1016/j.aim.2014.03.006.  Google Scholar

[13]

P. Petersen, Demystifying the Weitzenböck curvature operator Preprint available from: http://www.math.ucla.edu/~petersen/. Google Scholar

[14]

S. Raulot and A. Savo, On the first eigenvalue of the Dirichlet-to-Neumann operator on forms, J. of Functional Analysis, 262 (2012), 889-914.  doi: 10.1016/j.jfa.2011.10.008.  Google Scholar

[15]

G. Schwarz, Hodge Decomposition – A Method for Solving Boundary Value Problems, Lecture Notes in Math., Springer, (1995).  doi: 10.1007/BFb0095978.  Google Scholar

[16]

V. Sharafutdinov and C. Shonkwiler, The complete Dirichlet-to-Neumann map for differential forms, J. Geom. Anal., 23 (2013), 2063-2080.  doi: 10.1007/s12220-012-9320-6.  Google Scholar

[17]

Q. Wang and C. Xia, Sharp bounds for the first non-zero Stekloff eigenvalues, J. of Functional Analysis, 257 (2009), 2635-2644.  doi: 10.1016/j.jfa.2009.06.008.  Google Scholar

[18]

C. Xia, Rigidity for compact manifolds with boundary and non-negative Ricci curvature, Proc. Amer. Math. Soc., 125 (1997), 1801-1806.  doi: 10.1090/S0002-9939-97-04078-1.  Google Scholar

[19]

L. Yang and C. Yu, A higher dimensional generalization of Hersch-Payne-Schiffer inequality for Steklov eigenvalues, J. of Functional Analysis, 272 (2017), 4122-4130.  doi: 10.1016/j.jfa.2017.02.023.  Google Scholar

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