Foliated Plateau problems and asymptotic counting of surface subgroups

Sébastien Alvarez, Ben Lowe, Graham Smith

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the lower bound and the rigidity-at-equality via a precise equidistribution/foliated-Plateau framework (Theorem 1.1.1, Lemmas 4.5.1–4.5.3), including existence/uniqueness of k-disks and a foliation in the unit sphere bundle, then upgrades an area inequality to curvature rigidity (h ≡ −1) using measures on the bundle of marked k-disks. The candidate solution gets the right lower bound and the hyperbolic equality case by Gauss–Bonnet plus Kahn–Markovic counting, but its rigidity argument relies on unsubstantiated transfers of equidistribution across metrics and on foliation/“barrier” claims that the paper explicitly replaces with new machinery. In short, the model’s Step 5 is not justified, while the paper’s proof is complete (modulo a clear typographical ≥ vs > inconsistency in the statement).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper establishes a sharp asymptotic lower bound for counting quasi-Fuchsian surface subgroups in terms of areas of k-surfaces and proves curvature rigidity at equality. Its novel foliated-Plateau/equidistribution machinery circumvents obstacles faced by minimal-surface methods and appears robust for metrics with sectional curvature ≤ −1. The presentation is clear and self-contained; only minor editorial adjustments (inequality signs and a few clarifications) are needed.