An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917356656132096 |
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| author | Mukherjee, Mayukh |
| author_facet | Mukherjee, Mayukh |
| contents | A classical theorem of Colin de Verdière shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including multiplicities, subject to the usual topological constraints) by choosing a sufficiently heterogeneous smooth metric. In this paper, we study the same inverse problem under the rigid geometric constraint of \emph{constant negative sectional curvature}. Allowing the topological complexity to vary, we prove that any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature in any dimension $d\ge2$. In $d=2$ the construction uses hyperbolic collar degeneration and the discrete spectral limit theorems of Burger, building on the collar estimates of Buser; in $d\ge3$ we build macroscopically heterogeneous hyperbolic covering manifolds assembled from ``heavy'' vertex clusters and ``long'' corridor chains whose low-energy limit is a prescribed \emph{discrete} graph Laplacian. We also record the universal obstructions at curvature normalization $κ\equiv -1$: Yang-Yau in $d=2$ and Kazhdan-Margulis combined with Bishop--Gromov volume comparison in $d\ge3$. In particular, $λ_1$ is universally bounded at $κ=-1$, so target lists whose first positive eigenvalue exceeds this bound cannot be approximated within the class $κ\equiv -1$, and accommodating arbitrarily large prescribed $λ_1^*$ forces $|κ|\to\infty$. A corollary on the arbitrarily precise prescription of scale-invariant eigenvalue ratios at $κ\equiv -1$ and an explicit worked example are included. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21240 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature Mukherjee, Mayukh Spectral Theory Differential Geometry Geometric Topology A classical theorem of Colin de Verdière shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including multiplicities, subject to the usual topological constraints) by choosing a sufficiently heterogeneous smooth metric. In this paper, we study the same inverse problem under the rigid geometric constraint of \emph{constant negative sectional curvature}. Allowing the topological complexity to vary, we prove that any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature in any dimension $d\ge2$. In $d=2$ the construction uses hyperbolic collar degeneration and the discrete spectral limit theorems of Burger, building on the collar estimates of Buser; in $d\ge3$ we build macroscopically heterogeneous hyperbolic covering manifolds assembled from ``heavy'' vertex clusters and ``long'' corridor chains whose low-energy limit is a prescribed \emph{discrete} graph Laplacian. We also record the universal obstructions at curvature normalization $κ\equiv -1$: Yang-Yau in $d=2$ and Kazhdan-Margulis combined with Bishop--Gromov volume comparison in $d\ge3$. In particular, $λ_1$ is universally bounded at $κ=-1$, so target lists whose first positive eigenvalue exceeds this bound cannot be approximated within the class $κ\equiv -1$, and accommodating arbitrarily large prescribed $λ_1^*$ forces $|κ|\to\infty$. A corollary on the arbitrarily precise prescription of scale-invariant eigenvalue ratios at $κ\equiv -1$ and an explicit worked example are included. |
| title | An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature |
| topic | Spectral Theory Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2603.21240 |