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arxiv: 2606.12745 · v1 · pith:FKJXK5N7new · submitted 2026-06-10 · 🧮 math.DG · math.SP

A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

Xianzhe Dai , John Ennis , Xuan Hien Nguyen , Guofang Wei This is my paper

Pith reviewed 2026-06-27 08:01 UTC · model grok-4.3

classification 🧮 math.DG math.SP
keywords fundamental gaphoroconvex domainshyperbolic spaceDirichlet eigenvalueslarge diameterradial heightangular operator
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The pith

Horoconvex domains in hyperbolic space of curvature -1 obey a fundamental-gap lower bound of order D^{-3} for large diameter D.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the difference between the first two Dirichlet eigenvalues of a compact horoconvex domain in real hyperbolic space is bounded from below by a positive constant times D to the power -3 whenever the diameter D is large. The argument begins with a geometric reduction that replaces any such large domain by a model problem of fixed radial height and fixed width, and this reduction works uniformly in every dimension. The model problem is then solved by comparing the low-energy Dirichlet form against a limiting angular operator on the sphere while controlling the radial remainder with a one-dimensional spectral gap and endpoint Green-function estimates. A sympathetic reader cares because the fundamental gap governs the long-time decay rate of the heat kernel and the mixing time of Brownian motion on the domain.

Core claim

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature -1. The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial D^{-3} scale matching the known large-diameter upper bound.

What carries the argument

The geometric reduction of arbitrary large horoconvex domains to a fixed-width radial-height model problem, followed by Dirichlet-form comparison against a limiting angular operator on the sphere.

If this is right

  • The lower bound matches the scale of the existing upper bound, so the fundamental gap decays precisely like D^{-3}.
  • The dimension-independent geometric reduction implies the same polynomial rate holds in every dimension.
  • The separation of angular and radial contributions via the limiting operator and one-dimensional estimates controls the low-lying spectrum.
  • The same reduction technique applies directly to other eigenvalue problems on horoconvex domains of large diameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that the slowest-decaying mode is essentially determined by the angular ground state once the domain is thinned to fixed radial height.
  • One could test the sharpness by constructing families of horoconvex domains that achieve equality in the limit of the angular operator.
  • The method suggests that analogous gap bounds may hold for domains satisfying weaker convexity conditions provided a comparable reduction step can be found.

Load-bearing premise

The geometric reduction that converts arbitrary large horoconvex domains into a fixed-width radial-height model problem holds in all dimensions.

What would settle it

Numerical computation of the first two Dirichlet eigenvalues on a sequence of explicit horoconvex domains whose diameters tend to infinity, checking whether the observed gap remains at least on the order of D^{-3}.

Figures

Figures reproduced from arXiv: 2606.12745 by Guofang Wei, John Ennis, Xianzhe Dai, Xuan Hien Nguyen.

Figure 1
Figure 1. Figure 1: summarizes this representation. Radial-height representation Annular control h(θ) angle Radial-height profile angle height R h(θ) R − h(θ) Lipschitz [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen--Stancu--Wei large-diameter upper bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a lower bound of order D^{-3} for the fundamental gap of compact horoconvex domains with large diameter D in real hyperbolic space of curvature -1. The geometric part reduces arbitrary large horoconvex domains to a fixed-width radial-height model problem uniformly in all dimensions; the analytic part then establishes the model theorem by comparing the low-energy Dirichlet form against a limiting angular operator on the sphere, with the radial complement controlled by a one-dimensional branch gap and endpoint Green estimates.

Significance. If the central claim is correct, the result supplies the matching lower bound to the Nguyen--Stancu--Wei upper bound, establishing the sharp polynomial large-diameter scaling for the fundamental gap in the horoconvex setting. The uniformity of the geometric reduction across dimensions would constitute a substantive technical contribution to spectral geometry on hyperbolic manifolds.

major comments (2)
  1. [Geometric reduction (abstract and §3)] The geometric reduction (first step of the proof, as outlined in the abstract): the argument that this reduction maps the gap of an arbitrary large horoconvex domain onto the model gap without dimension-dependent losses or diameter-dependent errors that would degrade the claimed D^{-3} lower bound is load-bearing. Explicit constants and their independence of dimension must be verified for the polynomial scale to transfer.
  2. [Analytic part (abstract and §4)] Analytic comparison on the model (radial-height theorem): the separation of the low-energy Dirichlet form from the limiting angular operator via the one-dimensional branch gap and endpoint Green estimates must be shown to yield a lower bound whose constants remain uniform in the fixed-width parameter; any hidden dependence on the model width would prevent the D^{-3} scale from holding for the original domains.
minor comments (2)
  1. [Introduction and §2] Clarify the precise definition of horoconvexity employed (e.g., supporting horospheres, curvature conditions) and confirm it is used consistently in the reduction step.
  2. [Abstract and §1] The abstract states the result holds 'in all dimensions'; state the range of dimensions explicitly and note any dimension-dependent factors that appear in the estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed report. We appreciate the positive assessment of the significance of the result. We address each major comment below.

read point-by-point responses

  1. Referee: [Geometric reduction (abstract and §3)] The geometric reduction (first step of the proof, as outlined in the abstract): the argument that this reduction maps the gap of an arbitrary large horoconvex domain onto the model gap without dimension-dependent losses or diameter-dependent errors that would degrade the claimed D^{-3} lower bound is load-bearing. Explicit constants and their independence of dimension must be verified for the polynomial scale to transfer.

    Authors: The geometric reduction in Section 3 establishes a comparison between the fundamental gap of any large horoconvex domain and that of the fixed-width radial-height model. The estimates rely on the properties of horoconvexity in hyperbolic space, which provide uniform control independent of dimension. Specifically, the diameter-dependent errors are of lower order and do not affect the D^{-3} scaling. The constants in the inequalities are independent of the dimension n, as the proofs use comparison theorems that hold uniformly. We will add a remark in the revised version making the independence explicit by tracing the constants through the estimates. revision: yes

  2. Referee: [Analytic part (abstract and §4)] Analytic comparison on the model (radial-height theorem): the separation of the low-energy Dirichlet form from the limiting angular operator via the one-dimensional branch gap and endpoint Green estimates must be shown to yield a lower bound whose constants remain uniform in the fixed-width parameter; any hidden dependence on the model width would prevent the D^{-3} scale from holding for the original domains.

    Authors: In the analytic part, the model width is fixed independently of the domain and diameter D. The one-dimensional branch gap is positive and depends only on the fixed width, providing a uniform separation. The endpoint Green estimates are also uniform for the fixed model. Thus, the resulting lower bound constants are independent of the width parameter (as it is fixed) and of D, preserving the D^{-3} scale. No hidden dependence exists because the width is chosen fixed and the estimates are derived for that fixed case. We believe the current proof already establishes this uniformity, but we can add a sentence clarifying the independence from the width parameter. revision: partial

Circularity Check

0 steps flagged

Direct geometric reduction plus analytic comparison; derivation self-contained with no load-bearing self-citation or fitted-input renaming

full rationale

The abstract and outline present a two-part proof that is independent of its target bound: (1) a geometric reduction that converts arbitrary large-diameter horoconvex domains into a fixed-width radial-height model problem, stated to hold uniformly in all dimensions; (2) an analytic comparison on that model that pits the low-energy Dirichlet form against a limiting angular operator on the sphere, with separation supplied by a one-dimensional branch gap and endpoint Green estimates. The claimed D^{-3} lower bound is the output of these steps, not an input to them. The matching statement to the Nguyen–Stancu–Wei upper bound is merely a scale comparison after the fact and does not enter the derivation. No equations or citations are quoted that would reduce the lower bound to a self-definition, a fitted parameter, or a prior self-citation chain. This is the normal case of a self-contained argument against external geometric and analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of the Laplacian on hyperbolic space and the definition of horoconvexity; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Real hyperbolic space of curvature -1 admits a well-defined Laplacian and horoconvexity notion
    Background geometry invoked for the domain class and the eigenvalue problem.
  • domain assumption Large horoconvex domains reduce to fixed-width radial-height models
    Central geometric step stated in the abstract; if false the analytic comparison does not apply.

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