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arxiv: 2606.17697 · v1 · pith:NUUVPSODnew · submitted 2026-06-16 · 🧮 math.DG · math.AP

Sobolev and Michael-Simon inequalities via the ABP method beyond Euclidean volume growth

Debora Impera , Michele Rimoldi , Giona Veronelli This is my paper

Pith reviewed 2026-06-26 23:06 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords ABP methodMichael-Simon inequalitySobolev inequalityvolume noncollapsingnonnegative sectional curvatureNeumann potentialisoperimetric profileimmersed submanifolds
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The pith

A refinement of the ABP contact-set argument produces Michael-Simon inequalities with an explicit lower-order term on noncollapsing manifolds of nonnegative sectional curvature.

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

The paper develops an ABP approach to Sobolev and Michael-Simon inequalities that works under volume noncollapsing rather than requiring positive asymptotic volume ratio. The key step is showing that the ABP image of the contact set contains the entire geodesic ball of radius equal to the ABP parameter centered at the minimum of the Neumann potential. This containment lets the authors insert lower bounds on the volumes of geodesic balls at a fixed scale into the estimates. The resulting Michael-Simon inequality applies to immersed submanifolds with controlled mean curvature and includes a lower-order term scaled by the noncollapsing constant. In the intrinsic setting the same method recovers Varopoulos' L1-Sobolev inequality with an optimal gradient constant and gives explicit lower bounds on the isoperimetric profile.

Core claim

The central claim is that a refinement of Brendle's contact-set argument ensures the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential with radius equal to the ABP parameter. This geometric fact permits the direct insertion of volume lower bounds for geodesic balls, either at a fixed scale or under prescribed growth assumptions, yielding Michael-Simon type inequalities for immersed submanifolds in ambient manifolds with nonnegative sectional curvature and volume noncollapsing, as well as an ABP proof of Varopoulos' L1-Sobolev inequality with lower-order term and optimal constant in front of the gradient term.

What carries the argument

The refined contact-set argument in the ABP method, which guarantees that the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential with radius equal to the ABP parameter.

Load-bearing premise

The ABP image must contain the full geodesic ball centered at the minimum point of the Neumann potential with radius equal to the ABP parameter; without this containment the volume lower bounds cannot be inserted and the lower-order term disappears.

What would settle it

A concrete manifold with nonnegative sectional curvature and volume noncollapsing, together with a Neumann potential whose ABP image fails to contain the full geodesic ball of radius equal to the ABP parameter centered at its minimum, would show the refinement does not hold.

read the original abstract

We develop an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions. The main new observation is a refinement of Brendle's contact-set argument: the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This allows one to use lower bounds for the volumes of geodesic balls, either at a fixed scale or under prescribed volume-growth assumptions, rather than positive asymptotic volume ratio. The central application is a Michael-Simon type inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing. The resulting inequality contains a lower-order term determined by the noncollapsing scale and applies to submanifolds with controlled mean curvature. In the intrinsic case, the same method gives an ABP proof of Varopoulos' $L^{1}$-Sobolev inequality with lower-order term, identifying the optimal constant in front of the gradient term, as well as explicit lower bounds for the isoperimetric profile in terms of lower bounds on the volumes of geodesic balls. Further geometric applications include Topping-type diameter estimates for submanifolds involving the $L^{n-1}$-norm of the mean curvature and various heat kernel and spectral estimates in the minimal case.

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 develops an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions rather than positive asymptotic volume ratio. The central new observation is a refinement of Brendle's contact-set argument asserting that the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This permits insertion of lower bounds on volumes of geodesic balls at fixed scale. The main application is a Michael-Simon inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing, containing a lower-order term determined by the noncollapsing scale and valid for submanifolds with controlled mean curvature. Additional results include an ABP proof of Varopoulos' L^1-Sobolev inequality with lower-order term (identifying the optimal constant in front of the gradient term), explicit lower bounds for the isoperimetric profile, Topping-type diameter estimates, and heat-kernel/spectral estimates in the minimal case.

Significance. If the geometric containment holds, the work supplies a flexible extension of the ABP method to settings with only local volume noncollapsing, yielding inequalities with explicit dependence on the noncollapsing scale. The identification of the optimal gradient constant in the intrinsic Varopoulos inequality and the explicit isoperimetric-profile bounds are concrete strengths that could be useful for further geometric analysis.

major comments (2)
  1. [§3] §3 (refinement of contact-set argument): the claim that the ABP image contains the full geodesic ball of radius equal to the ABP parameter at the Neumann-potential minimum is load-bearing for recovering the lower-order term from fixed-scale volume bounds. The argument must still control the geometry of supporting hyperplanes under nonnegative sectional curvature; an explicit verification that the containment holds on a set of full measure (or a counter-example check) is required before the volume-insertion step can be used.
  2. [Theorem 1.1] Theorem 1.1 (Michael-Simon inequality): the lower-order term is stated to be determined by the noncollapsing scale, but the dependence on the mean-curvature bound and the precise radius of the geodesic ball must be tracked through the ABP estimate; without an explicit error term or constant, it is unclear whether the inequality reduces to the classical form when the noncollapsing scale tends to zero.
minor comments (2)
  1. [Introduction] The introduction should include a short comparison table or paragraph contrasting the new lower-order term with the classical Michael-Simon inequality under Euclidean volume growth.
  2. [§2 and later sections] Notation for the ABP parameter and the Neumann potential should be fixed once in §2 and used consistently; several later sections reuse the same symbols with slightly different normalizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive major comments. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (refinement of contact-set argument): the claim that the ABP image contains the full geodesic ball of radius equal to the ABP parameter at the Neumann-potential minimum is load-bearing for recovering the lower-order term from fixed-scale volume bounds. The argument must still control the geometry of supporting hyperplanes under nonnegative sectional curvature; an explicit verification that the containment holds on a set of full measure (or a counter-example check) is required before the volume-insertion step can be used.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will expand §3 with a detailed verification that the containment holds on a set of full measure. This will include a careful analysis of the supporting hyperplanes, using the nonnegative sectional curvature to control their geometry and confirm the required inclusion almost everywhere, prior to the volume-insertion step. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 (Michael-Simon inequality): the lower-order term is stated to be determined by the noncollapsing scale, but the dependence on the mean-curvature bound and the precise radius of the geodesic ball must be tracked through the ABP estimate; without an explicit error term or constant, it is unclear whether the inequality reduces to the classical form when the noncollapsing scale tends to zero.

    Authors: We acknowledge the need for explicit tracking. In the revision we will augment the statement and proof of Theorem 1.1 to record the precise dependence of all constants on the mean-curvature bound and the geodesic-ball radius appearing in the ABP estimate. We will also add a remark demonstrating that the lower-order term vanishes (recovering the classical Michael-Simon inequality) in the limit as the noncollapsing scale tends to zero. revision: yes

Circularity Check

0 steps flagged

No significant circularity: central claim rests on new geometric observation plus standard volume assumptions

full rationale

The paper's derivation chain begins with a claimed refinement of Brendle's contact-set argument (the ABP image contains the full geodesic ball of radius equal to the ABP parameter at the Neumann potential minimum). This is presented as the main new observation, which then permits inserting volume lower bounds from noncollapsing at fixed scale. No equations or steps reduce by construction to fitted parameters, self-definitions, or prior self-citations that themselves depend on the target inequality. The Michael-Simon application follows from this observation combined with nonnegative sectional curvature and volume noncollapsing, without renaming known results or smuggling ansatzes via self-citation. The argument is self-contained against external benchmarks (Brendle's prior work is cited as the base being refined, not as a load-bearing uniqueness theorem).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard differential-geometry background plus the new geometric containment claim; no free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption Riemannian manifolds with nonnegative sectional curvature admit the stated volume noncollapsing and geodesic-ball lower bounds
    Invoked for the central application to immersed submanifolds
  • standard math Standard properties of the ABP method and Neumann potential on manifolds
    Background for the contact-set argument

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