Spectral multiplicity and nodal sets for generic torus-invariant metrics

Donato Cianci; Chris Judge; Samuel Lin; Craig Sutton

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arXiv · 2207.14405

Spectral multiplicity and nodal sets for generic torus-invariant metrics

Donato Cianci·Chris Judge·Samuel Lin·Craig Sutton

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Abstract

Let a torus $T$ act freely on a closed manifold $M$ of dimension at least two. We demonstrate that, for a generic $T$-invariant Riemannian metric $g$ on $M$, each real $\Delta_g$-eigenspace is an irreducible real representation of $T$ and, therefore, has dimension at most two. We also show that, for the generic $T$-invariant metric on $M$, if $u$ is a non-invariant real-valued $\Delta_g$-eigenfunction that vanishes on some $T$-orbit, then the nodal set of $u$ is a connected smooth hypersurface whose complement has exactly two connected components.

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BibTeX RIS

Donato Cianci, Chris Judge, Samuel Lin, Craig Sutton. 2022-07-28. Spectral multiplicity and nodal sets for generic torus-invariant metrics. https://arxiv.org/abs/2207.14405

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