Pólya-Type Spectral Inequalities for Manifold Families

Definitions and mathematical statement

The intended research question is to develop Pólya-type lower eigenvalue inequalities, with suitable correction terms, uniformly over specified families of Riemannian manifolds. The supplied record does not specify those families, the spectral operator, boundary conditions, metric normalization, correction terms, or allowed dependence of constants. Consequently, it does not determine a fully quantified manifold conjecture.

The record also supplies the following Euclidean-domain conjectural bounds. Let $n$ denote the ambient dimension, let $|\Omega|$ denote Euclidean volume for $\Omega\subset{\mathbb{R}}^n$, and let $k\in{\mathbb{N}}=\{1,2,\ldots\}$. The notation $\rho_k(\Omega)$ denotes a spectral quantity whose definition is missing from the supplied record. Subject to resolving that omission and specifying the admissible domains, the stated quantifiers are: for every $\alpha>0$ and $M>0$, there exist $a=a(\alpha,M,n)>0$ and $b=b(\alpha,M,n)>0$ such that, for every admissible $\Omega\subset{\mathbb{R}}^n$ with $|\Omega|=M$ and every $k\in{\mathbb{N}}$, \[ \begin{aligned} \rho_k(\Omega)&\geq a\,k^{2/(2n-1)} &&\text{if $\Omega$ is convex},\\ \rho_k(\Omega)&\geq b\,k^{1/n} &&\text{for general admissible $\Omega$}. \end{aligned} \] These are transcribed conjectural bounds, not a complete mathematical formulation: the role of $\alpha$, including any implicit occurrence in $\rho_k$, and the hypotheses on $\Omega$ are unspecified.

The source motivates these bounds by results for rectangles and unions of rectangles attributed to its reference [5]. It also states that an underlying conjecture fails for manifolds in general, while results for round $n$-hemispheres $H_+^n$, attributed to its reference [6], allow an inequality of a similar type after adding a correction term. The antecedent of the claimed failure is unclear. The hemisphere formula is corrupted in the supplied extraction: it contains $\lambda_k$, $C_n$, a power $2/n$, and the apparent subtractive term $(n-1)(n-2)/6$, but their complete arrangement and normalization cannot be recovered reliably. No hemisphere inequality or universal manifold assertion is therefore imposed here.

Short English statement

Seek Pólya-type lower eigenvalue bounds with correction terms that hold uniformly over suitable families of Riemannian manifolds. The supplied Euclidean conjectures predict growth rates $k^{2/(2n-1)}$ for convex domains and $k^{1/n}$ for general domains of fixed volume, but the spectral quantity and admissibility conditions are missing. A precise final formulation requires these definitions and recovery of the corrupted hemisphere formula.

Sources

• Local UnsolvedMath registry, permanent ID $30005615$, title: Pólya-Type Spectral Inequalities for Manifold Families; exact source problem code: catalogue entry. The record gives proposed year 2023 and supplies no named proposer.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, pp. 2017--2106, DOI 10.4171/owr/2023/36. Source URL and supplied literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2023/36. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• The source extract attributes results for rectangles and unions of rectangles to reference [5], and results for round hemispheres with a correction term to reference [6]. Bibliographic details for these numbered references are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the bounds for $\rho_k$ as conjectures, and reports finding no general proof or counterexample. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims; this draft does not independently verify them or claim to have consulted the report.