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arxiv: 2606.13335 · v1 · pith:FVKNAINWnew · submitted 2026-06-11 · 🧮 math.DG · math.AP

Sharp stability of Alexandrov's theorem for C¹ domains in the small-excess regime

Alessio Figalli , Yi Ru-Ya Zhang This is my paper

Pith reviewed 2026-06-27 05:44 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords Alexandrov theoremquantitative stabilitymean curvatureC1 domainssets of finite perimetersmall excessstability estimates
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The pith

Bounded C1 sets with volume of the unit ball and small excess have excess and symmetric difference to the ball controlled by L2 oscillation of mean curvature.

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

The paper proves a sharp quantitative stability result for Alexandrov's theorem applying to bounded C1 open sets in any dimension. If such a set E has the same volume as the unit ball, sufficiently small excess, and distributional mean curvature in L2 on the boundary, then up to translation the excess of E, the measure of its symmetric difference with the ball, and any constant deviation of the curvature from n-1 are all bounded by a dimension-dependent multiple of the squared L2 norm of the curvature oscillation. The result gives control in a regime that does not assume the boundary is everywhere a graph over the sphere. All constants in the inequality depend only on dimension, while the C1 assumption serves only as a qualitative tool in the argument.

Core claim

If E subset R^n is a bounded C1 open set with the same volume as the unit ball B, small excess, and scalar distributional mean curvature H_∂E in L2(∂E), then up to a translation Exc(E) + |E Δ B|^2 + |μ - (n-1)|^2 ≤ C(n) ||H_∂E - μ||_L2(∂E)^2 for all real μ. In other words both the excess and the symmetric difference from the ball are controlled by the optimal L2 oscillation of the mean curvature. This yields a sharp stability estimate in a genuinely non-parametric regime.

What carries the argument

A BV version of Fuglede's spectral-gap argument combined with star-shaped rearrangement for sets of finite perimeter, quantitative tentacle estimates, and polyhedral approximation for non-graphical regions.

If this is right

  • Both geometric excess and symmetric difference with the ball are bounded by the L2 curvature oscillation.
  • The estimate holds for C1 boundaries without requiring higher regularity or global graphical structure.
  • All constants in the final bound depend only on dimension.
  • The C1 assumption is used only qualitatively and does not affect the dependence of the constants.

Where Pith is reading between the lines

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

  • The same combination of spectral-gap and rearrangement tools might apply to stability questions for other curvature quantities if an analogous gap can be established.
  • It would be natural to test whether the small-excess hypothesis can be dropped when stronger integrability on the curvature is assumed.
  • The result indicates that L2 measurements of mean curvature suffice to detect quantitative closeness to spheres when the set is already known to be close in excess.

Load-bearing premise

The excess must be small enough that the spectral-gap, rearrangement, tentacle, and approximation arguments close without large uncontrolled boundary pieces.

What would settle it

A sequence of bounded C1 sets with volume equal to the unit ball, excess tending to zero, but L2 curvature oscillation strictly smaller than a fixed multiple of the square root of the excess would violate the claimed inequality.

Figures

Figures reproduced from arXiv: 2606.13335 by Alessio Figalli, Yi Ru-Ya Zhang.

Figure 1
Figure 1. Figure 1: Planar picture for the boundary term |x| −nx · ν τ ∂F ∩∂H in Lemma 4.9. The radial side R of the layer Lm may meet ∂F at a point A in the interior of a side of the polygon that is not a curvature point. The term at A is bounded by ρ −1 A , while the term at the corner point B is bounded by −ρ −1 B plus the corner angle. Since ρA ≥ ρB, the non-curvature part is non-positive, while the corner error is contro… view at source ↗
read the original abstract

We prove a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension for bounded $C^1$ open sets in a small-excess regime. More precisely, if $E\subset \mathbb R^n$ is a bounded $C^1$ open set with the same volume as the unit ball $B$, small excess, and scalar distributional mean curvature $\mathcal H_{\partial E}\in L^2(\partial E)$, then, up to a translation, $$ \operatorname{Exc}(E)+|E\Delta B|^2+|\mu-(n-1)|^2 \le C(n)\|\mathcal H_{\partial E}-\mu\|_{L^2(\partial E)}^2 \qquad \forall\,\mu\in \mathbb R. $$ In other words, both the excess and the symmetric difference from the ball are controlled by the optimal $L^2$-oscillation of the mean curvature. This yields a sharp stability estimate in a genuinely non-parametric regime. The proof combines a $BV$ version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region. We note that the $C^1$ regularity assumption enters only as a qualitative technical ingredient of the proof, but all constants in the final estimate depend only on the dimension.

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

1 major / 0 minor

Summary. The paper proves a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension: for a bounded C^1 open set E ⊂ R^n with |E| = |B| (unit ball), small excess, and scalar distributional mean curvature H_∂E ∈ L^2(∂E), one has Exc(E) + |E Δ B|^2 + |μ − (n−1)|^2 ≤ C(n) ||H_∂E − μ||_{L^2(∂E)}^2 for all μ ∈ R (up to translation). The proof combines a BV version of Fuglede's spectral-gap argument, star-shaped rearrangement, quantitative tentacle estimates, and polyhedral approximation on the non-graphical region; C^1 regularity is used only qualitatively while all constants depend solely on n.

Significance. If the result holds, it yields the first sharp L^2-stability estimate for Alexandrov's theorem in a genuinely non-parametric (C^1) regime under a small-excess assumption. The combination of BV spectral-gap tools with rearrangement and approximation techniques is technically novel and produces a clean, dimension-dependent constant; the explicit control of both excess and symmetric difference by the optimal L^2 oscillation of mean curvature is a clear advance over prior quantitative Alexandrov results.

major comments (1)
  1. [polyhedral approximation argument (non-graphical region)] The polyhedral approximation argument for the non-graphical region (after star-shaped rearrangement and tentacle control) must be shown to produce an L^2 error for H_∂E that is bounded by a constant depending only on n, without implicit dependence on the C^1 modulus of the local graphs. The small-excess regime controls volume and perimeter but does not a priori give a uniform C^1 bound; if the number of polyhedral pieces or the approximation error in ||H_∂E − μ||_L2 grows with the modulus, the claimed C(n)-only dependence fails. This is load-bearing for the central inequality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on the polyhedral approximation step. We address the concern below and will revise the manuscript accordingly to make the n-only dependence fully explicit.

read point-by-point responses

  1. Referee: The polyhedral approximation argument for the non-graphical region (after star-shaped rearrangement and tentacle control) must be shown to produce an L^2 error for H_∂E that is bounded by a constant depending only on n, without implicit dependence on the C^1 modulus of the local graphs. The small-excess regime controls volume and perimeter but does not a priori give a uniform C^1 bound; if the number of polyhedral pieces or the approximation error in ||H_∂E − μ||_L2 grows with the modulus, the claimed C(n)-only dependence fails. This is load-bearing for the central inequality.

    Authors: The star-shaped rearrangement and quantitative tentacle estimates (both controlled by the small excess) first reduce the non-graphical region to a set whose total measure and perimeter are bounded by a constant depending only on n and the excess. This allows a covering by O(1) polyhedral pieces whose diameters and number are likewise controlled solely by n (via a Vitali-type covering argument that uses only the BV structure and the L^1 closeness to the sphere). The subsequent polyhedral approximation of the boundary is performed at a scale determined by the excess; the resulting L^2 error on H_∂E is estimated by integrating the difference against test functions whose gradients are bounded by n-dependent constants, using the L^2 integrability of H and the fact that the approximation error on each piece is absorbed into the excess term. Consequently the overall constant remains C(n). We will add a dedicated lemma (with full details of the covering and error estimates) in the revised version to make this independence from the C^1 modulus transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation combines external tools without reduction to inputs

full rationale

The provided abstract and description outline a proof that invokes a BV version of Fuglede's spectral-gap argument (external), star-shaped rearrangement for finite perimeter sets, tentacle estimates, and polyhedral approximation. The C1 assumption is stated to enter only qualitatively while constants depend solely on n. No equations or steps are quoted that reduce a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The central inequality is presented as a consequence of these standard techniques adapted to the small-excess regime, making the derivation self-contained against external mathematical results rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard tools from geometric measure theory; the small-excess regime is the key enabling assumption.

axioms (3)
  • standard math Properties of sets of finite perimeter and BV functions in R^n
    Invoked for the BV version of Fuglede's spectral-gap argument
  • domain assumption Existence of star-shaped rearrangement for sets of finite perimeter
    Used to reduce to a more graphical setting
  • domain assumption Quantitative estimates for tentacles and polyhedral approximation of non-graphical regions
    Required to control the boundary parts outside the star-shaped region

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The quantitative isoperimetric inequality: A calibration argument

    math.AP 2026-06 unverdicted novelty 7.0

    Short self-contained proof of the quantitative isoperimetric inequality via quantitative calibrations that control asymmetry and excess.

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