New Asymptotic Geometric Quantities in Riemannian Geometry and Their Geometric and Dynamical Applications
Pith reviewed 2026-05-10 10:35 UTC · model grok-4.3
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.
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
- 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
We introduce large $p$ asymptotic geometric quantities associated with $p$-capacity, the first $p$-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds. We prove the hierarchy $$ \mathcal{V}(M)\geq \mathcal C(\Omega)\geq \Lambda(M)=\mathcal M(M)\geq0, $$ and show that, under a centered-ball isoperimetric condition or a rotational symmetry condition, these quantities coincide with the volume entropy or the dimension. In the Hadamard nonpositively curved case it also agrees with the topological entropy of the geodesic flow. As an application, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also prove a second-order refinement. For a Hadamard manifold with compact quotient, under certain condition, the first-order large $p$ capacitary limit detects volume entropy, whereas the logarithmic second-order correction detects the rank.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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)
- [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.
- [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)
- [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.
- [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
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
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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
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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
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
axioms (2)
- domain assumption M is a complete noncompact Riemannian manifold
- domain assumption The large-p limits defining C(Ω), Λ(M), and M(M) exist
invented entities (3)
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infinity capacity C(Ω)
no independent evidence
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infinity eigenvalue Λ(M)
no independent evidence
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Maz'ya limit M(M)
no independent evidence
discussion (0)
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