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Niveau: Supérieur, Doctorat, Bac+8
Invent. math. 93, 161-183 (1988) Inventiones mathematicae 9 Springer-Verlag 1988 Nodal sets of eigenfunctions on Riemannian manifolds Harold Donnelly *' 1 and Charles Fefferman **' 2 Department of Mathematics, Purdue University, West Lafayette, IN47907, USA 2 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA 1. Introduction Let M be a compact connected manifold, with C ~ Riemannian metric. The Laplacian A of M is a negative definite, setf-adjoint, elliptic operator. Suppose that Fis a real eigenfunction of A with eigenvalue 2,A F = - 2F. The nodal set N of F is defined to be the set of points x ~ M where F(x)= 0. The unique continuation theorem \[1\] states that F never vanishes to infinite order. This places strong restrictions on the zeroes of F. By developing the machinery of Aronszajn \[1\], we establish a number of quantitative results concerning the nodal set. These theorems eem most interesting for large 2. One of our main conclusions is Theorem 1.1, The eigenfunction F vanishes at most to order c\[//2, for any point in M. When M is two dimensional, it follows from the work of Cheng \[5\], that F vanishes at most to order c2.