Generic properties of Laplace eigenfunctions in the presence of torus actions

Abstract: A result of Uhlenbeck (1976) states that for a generic Riemannian metric $g$ on a closed manifold $M$ of dimension at least two the real eigenspaces of the associated Laplace operator $\Delta_g$ are each one-dimensional and the nodal set (i.e., zero set) of any $\Delta_g$-eigenfunction is a smooth hypersurface. Now, let $T$ be a non-trivial torus acting freely on a closed manifold $M$ with $\dim M > \dim T$. We demonstrate that a generic $T$-invariant metric $g$ on $M$ has the following properties: (1) the real $\Delta_g$-eigenspaces are irreducible representations of $T$ and, consequently, are of dimension one or two, and (2) the nodal set of any $\Delta_g$-eigenfunction is a smooth hypersurface. The first of these statements is a mathematically rigorous instance of the belief in quantum mechanics that non-irreducible eigenspaces are ``accidental degeneracies.''

Regarding the second statement, in the event the non-trivial quotient $B = M/T$ satisfies a certain topological condition, we show that, for a generic $T$-invariant metric $g$, any orthonormal basis $\langle \phi_j \rangle$ consisting of $\Delta_g$-eigenfunctions possesses a density-one subsequence $\langle \phi_{j_k}\rangle$ where the nodal set of each $\phi_{j_k}$ is a smooth hypersurface dividing $M$ into exactly two nodal domains, the minimal possible number of nodal domains for a non-constant eigenfunction. This observation stands in stark contrast to the expected behavior of the nodal count in the presence of an ergodic geodesic flow, where examples suggest one should anticipate the nodal count associated to a ``typical'' sequence of orthogonal Laplace eigenfunctions will approach infinity.

This is joint work with Donato Cianci (GEICO), Chris Judge (Indiana) and Samuel Lin (Oklahoma).

differential geometryspectral theory

Audience: researchers in the topic

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