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The optimal eigenvalue count in Aubry's spectral sphere theorem for dimensions at least four

Anass Nifa

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Abstract

For every n≥4n\ge4, we determine the least number of positive Laplace eigenvalues whose pinching to nn, under Ricg≥(n−1)g\operatorname{Ric}_g\ge(n-1)g, forces a closed connected nn-manifold to be homeomorphic or diffeomorphic to SnS^n. This number is nn, also within the simply connected class. The examples are defined on a fixed simply connected manifold MnM_n with H2(Mn;ℤ)≅ℤ2H_2(M_n;\mathbb{Z})\cong\mathbb{Z}^2. At every sufficiently small prescribed volume V>0V>0, it admits a smooth family with λ1,…,λn−1→n\lambda_1,\ldots,\lambda_{n-1}\to n and λn≥n+cn\lambda_n\ge n+c_n, where cn>0c_n>0 is an Aubry constant. The proof uses the fixed-volume degeneration theorem for core-admissible doubles from a companion preprint. Applying that theorem also to SnS^n gives two families, on nonhomeomorphic manifolds, with the same volume, Ricci lower bound, and metric-measure limit Sn−2*S2πq1S^{n-2}*S^1_{2\pi q}, where q=V/Vol(Sn,gSn)q=V/\operatorname{Vol}(S^n,g_{S^n}). Spectral convergence gives the same limit for every fixed eigenvalue index. For q=1/mq=1/m, the known spectrum of the corresponding cyclic spherical quotient makes these limits explicit. In particular, for all sufficiently large integers m≥3m\ge3, the nnth positive eigenvalue of both families tends to 2(n+1)2(n+1). Each family has constant volume density and, for fixed VV, a uniform positive lower bound for the volumes of unit balls.

MSC 2020

58J50Primary
Spectral problems; spectral geometry; scattering theory on manifolds
53C20Secondary
Global Riemannian geometry, including pinching
53C21Secondary
Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23Secondary
Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Mathematics Subject Classification 2020

Record

Version DOI
10.5281/zenodo.22949597
All versions
10.5281/zenodo.22696869
Subjects
Spectral geometry; Ricci curvature
Research index
OpenAlex record
Language
English

Cite this paper

Anass Nifa. The optimal eigenvalue count in Aubry's spectral sphere theorem for dimensions at least four. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.22949597

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