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

Abstract

For every $n\ge4$, we determine the least number of positive Laplace eigenvalues whose pinching above the Lichnerowicz bound forces sphere topology under $\Ric_g\ge(n-1)g$. This number is $n$, for both homeomorphism and diffeomorphism, also in the simply connected class. At every sufficiently small prescribed positive volume $V$, a fixed simply connected nonspherical manifold with second homology $\Z^2$ admits a smooth family for which the first $n-1$ positive eigenvalues tend to $n$, while the $n$th remains uniformly separated from $n$. The geometric input is a fixed-volume degeneration of core-admissible doubles from a preceding paper. Applying it also to the sphere, and using measured spectral convergence, gives families on nonhomeomorphic manifolds, with the same fixed volume and Ricci lower bound, for which every scalar Laplace eigenvalue has the same limit. Their common metric-measure limit is $S^{n-2}*S^1_{2\pi q}$, where $q=V/\sigma_n$ and $\sigma_n=\Vol(S^n, g_{S^n})$. For $q=1/m$, $m\ge2$, we identify its canonical Laplacian with the invariant round-sphere Laplacian and compute all multiplicities:\[ \sum_{\ell\ge0}a_\ell t^\ell =\frac{1+t^m}{(1-t)^{n-1}(1-t^m)},\qquad \mu_\ell=\ell(\ell+n-1).\] When $q=1/m$ is sufficiently small and $m\ge3$, the $n$th positive eigenvalue tends to $2(n+1)$. All families have constant volume density and a uniform positive lower bound on the volumes of unit balls.