11 February 2022
A Local version of Courant’s Nodal Domain Theorem
Let (M^n, g) denote a smooth and compact Riemannian manifold with no boundary equipped with a smooth Riemannian metric g. Courant’s nodal domain theorem asserts that for the Laplace-Beltrami operator on M, if we order the eigenvalues in increasing order with multiplicity, then the eigenfunction for the k-th eigenvalue has at most k connected components(the nodal domains) where the eigenfunction does not vanish. C. Fefferman and H. Donnelley proposed about 30 years ago a local version of this result on every ball in M. This local question is connected with the question of S.-T. Yau on the length of the zero set of the eigenfunctions. We propose to give answers to this question. This work is joint with A. Logunov E. Mallinikova and D. Mangoubi.