an:04078332
Zbl 0659.58047
Donnelly, Harold; Fefferman, Charles
Nodal sets of eigenfunctions on Riemannian manifolds
EN
Invent. Math. 93, No. 1, 161-183 (1988).
00168209
1988
j
58J50 53C20
Laplacian; eigenfunction; nodal set
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, \(c_ 1\sqrt{\lambda}\leq {\mathcal H}^{n-1}N\leq c_ 2\sqrt{\lambda}\).
H.Donnelly