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Bass note spectrum of compact hyperbolic 2-orbifolds

Characterize the set of possible first Laplace eigenvalues λ1 of compact hyperbolic 2-orbifolds: prove that the bass note spectrum is exactly (0, 15.7902...] ∪ {23.0785...} ∪ {28.0798...} ∪ {44.8883...}, corresponding to specific triangle orbifolds.

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Background

The authors summarize a conjecture (originating from Kravchuk–Mazáč–Pal) describing the full set of λ1 values attainable by compact hyperbolic 2-orbifolds: an interval together with three singleton values corresponding to particular triangle orbifolds.

They present new bounds consistent with this picture and discuss how their hyperbolic bootstrap framework connects to and supports such spectral characterizations, while noting that a complete proof remains conjectural.

References

\begin{conj} \label{conj:bass} The bass note spectrum of compact hyperbolic 2-orbifolds is \begin{align} \label{eqn:bass_note_spectrum_conj} (0,15.7902...] \cup {23.0785...} \cup {28.0798...} \cup {44.8883...}, \end{align} where these four numbers are the first eigenvalues of specific triangle orbifolds given in Conjecture~4.2. \end{conj}

— A converse theorem for hyperbolic surface spectra and the conformal bootstrap (2509.17935 - Adve, 22 Sep 2025) in Conjecture 6.1, Section 6.1 (Bounds on λ1)