Luigi Provenzano: On the critical points of Steklov eigenfunctions

(22 avril 2024/April 22, 2024) Seminar Spectral Geometry in the clouds
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Luigi Provenzano (Università di Roma): On the critical points of Steklov eigenfunctions
Abstract: We consider the critical points of Steklov eigenfunctions on a compact, smooth n-dimensional Riemannian manifold M with boundary ∂M. For generic metrics on M we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to ∂M, and the Euler characteristic of M. In dimension 2 this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of M and of the number of sign changes of u on ∂M. Based on a joint work with Luca Battaglia (Universit`a degli Studi Roma Tre) and Angela Pistoia (Sapienza Universit`a di Roma).