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Nodal sets of eigenfunctions on Riemannian manifolds. (English) Zbl 0659.58047
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

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
Full Text: DOI EuDML
[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) · Zbl 0358.58015 · 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
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