Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula

Bonifacio, James; Mazáč, Dalimil; Pal, Sridip

doi:10.1007/s00220-024-05222-0

Title: Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula

Journal Article · 16 February 2025 · Communications in Mathematical Physics

DOI: https://doi.org/10.1007/s00220-024-05222-0 · OSTI ID:2519266

^[1]; Mazáč, Dalimil ^[2]; Pal, Sridip ^[3]

University of Mississippi, MS (United States)
Institute for Advanced Study, Princeton, NJ (United States)
Institute for Advanced Study, Princeton, NJ (United States); California Institute of Technology, Pasadena, CA (United States)

We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace--Beltrami operator on functions and the Laplacian on powers of the cotangent bundle. Our approach employs linear programming techniques to derive rigorous bounds by leveraging two types of spectral identities. The first type, inspired by the conformal bootstrap, arises from the consistency of the spectral decomposition of the product of Laplace eigensections, and involves the Laplacian spectra as well as integrals of triple products of eigensections. We formulate these conditions in the language of representation theory of PSL₂ (C) and use them to prove upper bounds on the first and second Laplacian eigenvalues. The second type of spectral identities follows from the Selberg trace formula. We use them to find upper bounds on the spectral gap of the Laplace--Beltrami operator on hyperbolic 3-orbifolds, as well as on the systole length of hyperbolic 3-manifolds, as a function of the volume. Further, we prove that the spectral gap λ₁ of the Laplace--Beltrami operator on all closed hyperbolic 3-manifolds satisfies λ₁ < 47.32. Along the way, we use the trace formula to estimate the low-energy spectra of a large set of example orbifolds and compare them with our general bounds, finding that the bounds are nearly sharp in several cases.

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Research Organization:: California Institute of Technology, Pasadena, CA (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC), High Energy Physics (HEP)

Grant/Contract Number:: SC0009988; SC0011632

OSTI ID:: 2519266

Journal Information:: Communications in Mathematical Physics, Journal Name: Communications in Mathematical Physics Journal Issue: 3 Vol. 406; ISSN 0010-3616; ISSN 1432-0916

Publisher:: Springer Science and Business Media LLCCopyright Statement

Country of Publication:: United States

Language:: English

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