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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
Keywords:
Laplacian; eigenfunction; nodal set
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References:
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