The APDE seminar on Monday, 9/18, will be given by Junehyuk Jung (Brown) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Nodal domains of equivariant eigenfunctions on Kaluza-Klein 3-folds.

Abstract: In this talk, I’m going to present my work with Steve Zelditch, where we prove that, when M is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically 2, independent of the eigenvalues. Note that principle S1-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when (M,g) is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to +∞. I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. In particular, this tells us that the number of nodal domain could be uniformly bounded independent of the eigenvalue.