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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

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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