Abstract
Let e λ (x) be an eigenfunction with respect to the Laplace-Beltrami operator Δ M on a compact Riemannian manifold M without boundary: Δ M e λ = λ 2 e λ . We show the following gradient estimate of e λ : for every λ ≥ 1, there holds \({\lambda\|e_\lambda\|_\infty/C\leq \|\nabla e_\lambda\|_\infty\leq C\lambda\|e_\lambda\|_\infty}\), where C is a positive constant depending only on M.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brüning J.: Über knoten von eigenfunktionen des Laplace-Beltrami operators. Math. Z. 158, 15–21 (1975)
Donnelly H., Fefferman C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)
Donnelly, H., Fefferman, C.: Growth and geometry of eigenfunctions of the Laplacian. Analysis and partial differential equations, Lecture Notes in Pure and Applied Math, vol. 122, pp. 635–655. Dekker, NY (1990)
Gilbarg, D., Trudinger, Neil S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin (2001)
Grieser D.: Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Commun. Partial Differential Equ. 27(7-8), 1283–1299 (2002)
Hörmander L.: The spectral function of an elliptic operator. Acta Math. 88, 341–370 (1968)
Mangoubi D.: Local asymmetry and the inner radius of nodal domain. Comm. Partial Differential Equ. 33(7-9), 1611–1621 (2008)
Nadirashvili N.S.: Metric properties of eigenfunctions of the Laplace operator on manifolds. Ann. Inst. Fourier (Grenoble) 41(1), 259–265 (1991)
Safarov Y., Vassiliev D.: The Asymptotic Distribution of Eigenvalues of Partial Differential Operators Translations of Mathematical Monograph, 155. American Mathematical Society, Providence, RI (1997)
Seeger A., Sogge C.D.: On the boundedness of functions of pseudo-differential operators on a compact manifolds. Duke Math. J. 59, 709–736 (1989)
Sogge C.D.: On the convergence of Riesz means of compact manifolds. Ann. Math. 126, 439–447 (1987)
Sogge C.D.: Eigenfunction and Bochner-Riesz estimates on manifolds with boundary. Math. Res. Lett. 9, 205–216 (2002)
Xu B.: Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold. Ann. Glob. Anal. Geom. 26(3), 231–252 (2004)
Xu, X.: Eigenfunction estimates on compact manifolds with boundary and Hörmander multiplier theorem. Ph. D. Thesis, Johns Hopkins University (2004)
Xu X.: New proof of the Hörmander multiplier theorem on compact manifolds without boundary. Proc. Am. Math. Soc. 135(5), 1585–1595 (2007)
Xu X.: Gradient estimates for eigenfunctions of compact manifolds with boundary and the Hörmander multiplier theorem. Forum Math. 21(3), 455–476 (2009)
Zelditch, S.: Local and global analysis of eigenfunctions on Riemannian manifolds. In: Handbook of Geometric Analysis, no. 1, 545–658, Adv. Lect. Math, vol 7. Int. Press, Somerville, MA (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shi, Y., Xu, B. Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary. Ann Glob Anal Geom 38, 21–26 (2010). https://doi.org/10.1007/s10455-010-9198-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-010-9198-0