Paper · Dec 12, 2023
The spectral rigidity of Ricci soliton and Einstein-type manifolds
As on arXiv
Abstract
We are concerned in this article with a classical topic in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of sectional curvature (resp. holomorphic sectional curvature) of a compact Riemannian manifold (resp. Kähler manifold) can be completely determined by the eigenvalues of its -Laplacian for a single integer ? We treat this question under two conditions: gradient shrinking Ricci soliton for Riemannian manifolds and cohomologically Einstein for Kähler manifolds. We show that, with some sporadic unknown cases, this is true for each . Furthermore, we show that the condition of being isospectral can be relaxed to a suitable almost-isospectral version.
- Submitted
- Dec 12, 2023
- Subjects
- math.DG, math.SP
Comments: 21 pages. arXiv admin note: text overlap with arXiv:1804.00517
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Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.