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- Title
Spectral Multiplicity and Nodal Domains of Torus-Invariant Metrics.
- Authors
Cianci, Donato; Judge, Chris; Lin, Samuel; Sutton, Craig
- Abstract
Let a |$d$| -dimensional torus |$\mathbb{T}$| act freely and smoothly on a closed manifold |$M$| of dimension |$n>d$|. We show that, for a generic |$\mathbb{T}$| -invariant Riemannian metric |$g$| on |$M$| , each real |$\Delta _{g}$| -eigenspace is an irreducible real representation of |$\mathbb{T}$| and, therefore, has dimension at most two. We also show that, for the generic |$\mathbb{T}$| -invariant metric |$g$| on |$M$| , if |$u$| is a non-invariant real-valued |$\Delta _{g}$| -eigenfunction that vanishes on some |$\mathbb{T}$| -orbit, then the nodal set of |$u$| is a connected smooth hypersurface. If |$n>d+1$| , we show that the complement of the nodal set has exactly two connected components. As a consequence, we obtain new examples of manifolds for which—up to a sequence of Weyl density zero—each eigenfunction has exactly two nodal domains.
- Subjects
RIEMANNIAN metric; MULTIPLICITY (Mathematics); EIGENFUNCTIONS; TORUS; HYPERSURFACES
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 3, p2192
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad102