A Local version of Courant's Nodal domain Theorem

Abstract: Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding to the k-th eigenvalue. It is not difficult to show that the number of zeros of u_k is exactly k+1. Equivalently, the number of connected components of the complement of $u_k=0$ is $k$.

In 1923 Courant found that in higher dimensions (considering eigenfunctions of the Laplacian on a closed Riemannian manifold M) the number of connected components of the open set $M\setminus {u_k=0}$ is at most $k$.

In 1988 Donnelly and Fefferman gave a bound on the number of connected components of $B\setminus {u_k=0}$, where $B$ is a ball in $M$. However, their estimate was not sharp (even for spherical harmonics).

We describe the ideas which give the sharp bound on the number of connected components in a ball. The talk is based on a joint work with S. Chanillo, A. Logunov and E. Malinnikova, with a contribution due to F. Nazarov.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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