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arxiv: 2604.14600 · v1 · submitted 2026-04-16 · 🧮 math.DG

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Relations of Four Asymptotic Geometric Quantities in Riemannian Geometry

Xiaoshang Jin , Jiabin Yin
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:35 UTC · model grok-4.3

classification 🧮 math.DG
keywords Riemannian manifoldsvolume entropyp-capacityDirichlet eigenvalueMaz'ya constantlarge p asymptoticsnoncompact geometry
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The pith

On complete noncompact Riemannian manifolds, volume entropy bounds infinity capacity, which bounds the infinity eigenvalue equal to the Maz'ya limit.

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

The paper examines the large-p limits of the p-capacity of compact sets, the first Dirichlet p-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds. It defines the corresponding infinity versions and proves that for any compact subset Ω the volume entropy V(M) is at least the infinity capacity C(Ω), which is at least the infinity eigenvalue Λ(M), which equals the Maz'ya limit M(M). A reader would care because the relations give a unified chain of inequalities that become equalities under isoperimetric control, rotational symmetry, or curvature bounds, and the paper supplies examples where the inequalities are strict.

Core claim

The paper establishes the general inequality V(M) ≥ C(Ω) ≥ Λ(M) = M(M) for any compact Ω subset M on a complete noncompact Riemannian manifold M, where V(M) denotes volume entropy, C(Ω) the infinity capacity of Ω, Λ(M) the infinity eigenvalue, and M(M) the Maz'ya limit. Under additional geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, all four quantities coincide and equal either V(M) or the dimension of M. Explicit examples demonstrate that strict inequality can hold in the absence of these conditions.

What carries the argument

The large-p asymptotic limits that define the infinity capacity C(Ω), infinity eigenvalue Λ(M), and Maz'ya limit M(M), together with the chain of inequalities that relates them to volume entropy.

If this is right

  • Under isoperimetric control of balls or curvature bounds, the infinity capacity, infinity eigenvalue, and Maz'ya limit all equal the volume entropy.
  • On rotationally symmetric manifolds the quantities coincide and equal either the volume entropy or the manifold dimension.
  • Strict inequality between the quantities is possible, as shown by concrete examples in the paper.
  • The equality of the infinity eigenvalue and Maz'ya limit holds in full generality without extra assumptions.

Where Pith is reading between the lines

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

  • The chain might let results proved for one quantity transfer directly to the others on noncompact manifolds.
  • Explicit calculations on model spaces such as hyperbolic space could confirm equality cases and quantify the gap when strict inequality holds.
  • The unification suggests these limits capture the same large-scale geometric information at infinity.

Load-bearing premise

The large-p limits that define the infinity capacity, infinity eigenvalue, and Maz'ya limit exist and are finite on complete noncompact Riemannian manifolds.

What would settle it

An explicit computation or counterexample manifold on which the infinity eigenvalue differs from the Maz'ya limit or on which the infinity capacity exceeds the volume entropy.

read the original abstract

This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(\Omega)$, the infinity eigenvalue $\Lambda(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $\Omega\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(\Omega) \ge \Lambda(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. We also provide examples showing strict inequalities hold.

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 paper introduces the infinity capacity C(Ω) of a compact set Ω, the infinity eigenvalue Λ(M), and the Maz'ya limit M(M) as large-p limits of the corresponding p-capacity, first Dirichlet p-eigenvalue, and Maz'ya constant on a complete noncompact Riemannian manifold M. It proves the general inequality V(M) ≥ C(Ω) ≥ Λ(M) = M(M), where V(M) is the volume entropy, for any compact Ω ⊂ M. Under additional assumptions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal V(M) or the dimension n; examples are given where the inequalities are strict.

Significance. If the large-p limits are shown to exist and the inequality holds without further hypotheses, the result would unify several asymptotic invariants tied to volume growth and provide a new tool for analyzing noncompact manifolds. The equality cases under geometric conditions and the strict-inequality examples strengthen the contribution, but the absence of existence guarantees limits the scope.

major comments (2)
  1. [Abstract] Abstract: The statement of a 'general inequality' V(M) ≥ C(Ω) ≥ Λ(M) = M(M) for arbitrary complete noncompact M presupposes that the large-p limits defining C(Ω), Λ(M), and M(M) exist in the extended reals. No curvature, volume-growth, or compactness hypotheses are stated to guarantee finiteness or existence, rendering the chain of inequalities formally undefined on some manifolds.
  2. [Definitions section] Definitions of the new quantities (presumably §2–3): The infinity eigenvalue Λ(M) is obtained as the p → ∞ limit of the first Dirichlet p-eigenvalue via the Rayleigh quotient on compactly supported functions, and the Maz'ya limit M(M) is defined analogously; without explicit conditions ensuring these limits are finite, the claimed equality Λ(M) = M(M) cannot be asserted in general.
minor comments (2)
  1. [Throughout] Notation: The script letters C, Λ, M, V are introduced for the new limits; a brief comparison table with the classical p-versions would improve readability.
  2. [Examples section] Examples: The strict-inequality examples are mentioned but their manifolds and computations are not detailed in the abstract; explicit coordinate expressions or curvature values would clarify the gap between the quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for raising the important question of existence of the large-p limits. We address each major comment below and will revise the manuscript to make the definitions fully rigorous in the extended reals.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement of a 'general inequality' V(M) ≥ C(Ω) ≥ Λ(M) = M(M) for arbitrary complete noncompact M presupposes that the large-p limits defining C(Ω), Λ(M), and M(M) exist in the extended reals. No curvature, volume-growth, or compactness hypotheses are stated to guarantee finiteness or existence, rendering the chain of inequalities formally undefined on some manifolds.

    Authors: We agree that the abstract should be more precise. In the paper the three quantities are defined via limsup as p → ∞ of the corresponding p-objects (which always exists in [0, +∞]). The stated chain of inequalities is proved for these limsup values. We will revise the abstract to read: 'We introduce the infinity capacity C(Ω), the infinity eigenvalue Λ(M), and the Maz'ya limit M(M) as the limsup as p → ∞ of the corresponding p-quantities, and establish the general inequality V(M) ≥ C(Ω) ≥ Λ(M) = M(M) in the extended reals.' revision: yes

  2. Referee: [Definitions section] Definitions of the new quantities (presumably §2–3): The infinity eigenvalue Λ(M) is obtained as the p → ∞ limit of the first Dirichlet p-eigenvalue via the Rayleigh quotient on compactly supported functions, and the Maz'ya limit M(M) is defined analogously; without explicit conditions ensuring these limits are finite, the claimed equality Λ(M) = M(M) cannot be asserted in general.

    Authors: The equality Λ(M) = M(M) is established by showing that the limsup of the first Dirichlet p-eigenvalue equals the limsup of the Maz'ya constant through direct comparison of their Rayleigh quotients; the argument holds verbatim when both limsups are +∞. We will insert a short paragraph at the beginning of the definitions section stating that all three quantities are defined as limsups (hence always exist in the extended reals) and that the equality is understood in this sense. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions and inequality are independently established

full rationale

The paper introduces infinity capacity C(Ω), infinity eigenvalue Λ(M), and Maz'ya limit M(M) as large-p limits of the standard p-capacity, first Dirichlet p-eigenvalue, and Maz'ya constant respectively. It then states and proves the chain V(M) ≥ C(Ω) ≥ Λ(M) = M(M) for any compact Ω ⊂ M, with V(M) the volume entropy. These are presented as separately defined quantities whose relations are derived from analysis on complete noncompact Riemannian manifolds, under additional geometric conditions for equality cases. No equation or step reduces one quantity to another by definition, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the paper. The equality Λ(M) = M(M) is asserted as a theorem result rather than an identity by construction. The existence of the limits is an assumption for the statements to make sense, but that is a well-posedness issue, not a circular reduction in the derivation. The central claim therefore retains independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The central claim rests on the existence of the large-p limits for capacity, eigenvalue, and Maz'ya constant on complete noncompact manifolds, plus standard Riemannian geometry axioms. No free parameters are introduced. Three new entities are defined.

axioms (2)
  • domain assumption M is a complete noncompact Riemannian manifold
    Stated in the abstract as the setting for all quantities.
  • domain assumption The large-p limits defining C(Ω), Λ(M), and M(M) exist
    Implicit in the introduction of the infinity quantities.
invented entities (3)
  • infinity capacity C(Ω) no independent evidence
    purpose: Large-p limit of p-capacity of compact set Ω
    New object introduced to relate to the other quantities.
  • infinity eigenvalue Λ(M) no independent evidence
    purpose: Large-p limit of first Dirichlet p-eigenvalue
    New limiting object in the inequality chain.
  • Maz'ya limit M(M) no independent evidence
    purpose: Large-p limit of Maz'ya constant
    New limiting object shown equal to Λ(M).

pith-pipeline@v0.9.0 · 5452 in / 1435 out tokens · 22313 ms · 2026-05-10T10:35:53.462837+00:00 · methodology

discussion (0)

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Reference graph

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