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Invent math (2011) 185:199237 DOI 10.1007/s00222-010-0307-2
On the nodal sets of toral eigenfunctions
Jean Bourgain Zev Rudnick
Received: 12 March 2010 / Accepted: 1 December 2010 / Published online: 22 December 2010 Springer-Verlag 2010
Abstract We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional at torus. The question we address is: Can a xed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues. Our main result is that this cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature.
In dimension two, the result follows from a uniform lower bound for the L2-norm of the restriction of eigenfunctions to the curve, proved in an earlier paper (Bourgain and Rudnick in C. R. Math. 347(2122):12491253, 2009). In high dimensions we currently do not have this bound. Instead, we make use of the real-analytic nature of the at torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain. In all of our results, we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere. We also present an independent proof for the two-dimensional case relying on the abc-theorem in function elds.
J. BourgainSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA e-mail: mailto:[email protected]
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Z. Rudnick ( )
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israele-mail: mailto:[email protected]
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200 J. Bourgain, Z. Rudnick
Contents
1 Introduction and statement of results . . . . . . . . . . . . . . . . . . 2001.1 Dimension d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . 2011.3 About the proofs . . . . . . . . . . . . . . ....