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Linear programming bounds for hyperbolic surfaces

Maxime Fortier Bourque, Bram Petri

TL;DR

This work extends linear programming bounds, originally developed for sphere packings, to closed hyperbolic surfaces by leveraging the Selberg trace formula and carefully constructed test functions. The authors derive rigorous upper bounds for five geometric/spectral invariants—sys, kiss, $\lambda_1$, $m_1$, and $N_{\mathrm{small}}$—valid across genus, with strong performance in low genus and robust asymptotics as genus grows; they also obtain lower bounds on systole that yield spectral gaps $>1/4$ in several genera. The approach unifies the bounds via a common LP framework, combining algebraic (polynomial) and Bessel-function-based test functions, interval arithmetic, and ancillary computational proofs to certify results. The results advance the understanding of optimal geometric and spectral configurations on hyperbolic surfaces and provide near-term pathways toward global bounds and Ramanujan-like phenomena across genera. Practical impact includes sharper universal inequalities and a framework for certifiable numerical proofs in spectral geometry.

Abstract

We adapt linear programming methods from sphere packings to closed hyperbolic surfaces and obtain new upper bounds on their systole, their kissing number, the first positive eigenvalue of their Laplacian, the multiplicity of their first eigenvalue, and their number of small eigenvalues. Apart from a few exceptions, the resulting bounds are the current best known both in low genus and as the genus tends to infinity. Our methods also provide lower bounds on the systole (achieved in genus $2$ to $7$, $14$, and $17$) that are sufficient for surfaces to have a spectral gap larger than $1/4$.

Linear programming bounds for hyperbolic surfaces

TL;DR

This work extends linear programming bounds, originally developed for sphere packings, to closed hyperbolic surfaces by leveraging the Selberg trace formula and carefully constructed test functions. The authors derive rigorous upper bounds for five geometric/spectral invariants—sys, kiss, , , and —valid across genus, with strong performance in low genus and robust asymptotics as genus grows; they also obtain lower bounds on systole that yield spectral gaps in several genera. The approach unifies the bounds via a common LP framework, combining algebraic (polynomial) and Bessel-function-based test functions, interval arithmetic, and ancillary computational proofs to certify results. The results advance the understanding of optimal geometric and spectral configurations on hyperbolic surfaces and provide near-term pathways toward global bounds and Ramanujan-like phenomena across genera. Practical impact includes sharper universal inequalities and a framework for certifiable numerical proofs in spectral geometry.

Abstract

We adapt linear programming methods from sphere packings to closed hyperbolic surfaces and obtain new upper bounds on their systole, their kissing number, the first positive eigenvalue of their Laplacian, the multiplicity of their first eigenvalue, and their number of small eigenvalues. Apart from a few exceptions, the resulting bounds are the current best known both in low genus and as the genus tends to infinity. Our methods also provide lower bounds on the systole (achieved in genus to , , and ) that are sufficient for surfaces to have a spectral gap larger than .
Paper Structure (29 sections, 25 theorems, 190 equations, 2 figures, 7 tables)

This paper contains 29 sections, 25 theorems, 190 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Fourier transform
  3. The Selberg trace formula
  4. Linear programming
  5. Certifying inequalities on intervals
  6. Certifying error bounds on integrals
  7. Ancillary files
  8. Bessel functions
  9. Systole
  10. The criterion
  11. Low genus
  12. Asymptotics
  13. Kissing numbers
  14. The criterion
  15. Low genus
  16. ...and 14 more sections

Key Result

Theorem \ref{thm:systole_asymp}

There exists some $g_0 \geq 2$ such that every closed hyperbolic surface $M$ of genus $g\geq g_0$ satisfies

Figures (2)

  • Figure 1: Upper bounds and current record holders for the maximization of geometric and spectral invariants associated to hyperbolic surfaces.
  • Figure 2: Upper bounds on the number of small eigenvalues and lower bounds on the systole that imply a spectral gap larger than a quarter.

Theorems & Definitions (60)

  • Theorem \ref{thm:systole_asymp}
  • Theorem \ref{thm:kiss_asymp}
  • Theorem \ref{thm:lamb_asymp}
  • Theorem \ref{thm:mult_asymp}
  • Theorem \ref{thm:small_asymp}
  • Theorem 6.1
  • proof
  • Remark 6.2
  • Remark 6.3
  • Theorem 6.4
  • ...and 50 more