Donnelly, Harold; Fefferman, Charles Nodal sets of eigenfunctions on Riemannian manifolds. (English) Zbl 0659.58047 Invent. Math. 93, No. 1, 161-183 (1988). Let \(\Delta\) denote the Laplacian of a compact connected Riemannian manifold M. Suppose that F is a real eigenfunction of \(\Delta\) with eigenvalue \(\lambda\). It is proved that F vanishes to at most order \(c\sqrt{\lambda}\), for any point in M. The nodal set N of F is defined to be the set of points where F vanishes. If M is real analytic, upper and lower bounds are obtained for the n-1-dimensional Hausdorff measure of N. More specifically, \(c_ 1\sqrt{\lambda}\leq {\mathcal H}^{n-1}N\leq c_ 2\sqrt{\lambda}\). Reviewer: H.Donnelly Cited in 14 ReviewsCited in 207 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching Keywords:Laplacian; eigenfunction; nodal set × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations of second order. J. Math. Pures Appl.36, 235-249 (1957) · Zbl 0084.30402 [2] Bishop, R., Crittenden, R.: Geometry of Manifolds. New York-London: Academic Press, 1964 · Zbl 0132.16003 [3] Brüning, J.: Über Knoten von Eigenfunktionen des Laplace Beltrami operator. Math. Z.158, 15-21 (1978) · doi:10.1007/BF01214561 [4] Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom.17, 15-53 (1982) · Zbl 0493.53035 [5] Cheng, S. Y.: Eigenfunctions and nodal sets. Comment. Math. Helv.51, 43-55 (1976) · Zbl 0334.35022 · doi:10.1007/BF02568142 [6] Cheng, S. Y., Yau, S. T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math.28, 333-354 (1975) · Zbl 0312.53031 · doi:10.1002/cpa.3160280303 [7] Federer, H.: Geometric measure, theory. Berlin-Heidelberg-New York: Springer 1969 · Zbl 0176.00801 [8] Hörmander, L.: Linear partial differential operators. Berlin-Heidelberg-New York: Springer 1963 · Zbl 0108.09301 [9] Stein, E., Weiss, G.: Fourier analysis on Euclidean spaces. Princeton: Princeton University Press 1971 · Zbl 0232.42007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.