BIRS Workshop Lecture Videos
Boundedness of the number of nodal domains of eigenfunctions Jung, Junehyuk
The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with non-empty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty. In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic: for any given orthonormal eigenbasis one can find a subsequence of density 1 where the number of nodal domains is identically 2. This highlights how underlying dynamics can impact the nodal counting. I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1.Â I also will present an explicit orthonormal eigenbasis on the 3 torus where all of them have only two nodal domains.
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