Ciraolo, Figalli, Maggi and Novaga (Crelle 741, 2018) proved that a bounded set in Rn\mathbb{R}^n whose nonlocal mean curvature of order s∈(0,12)s\in(0,\tfrac12) is constant must be a ball, and established sharp quantitative stability: the scale-invariant distance of the set from a ball, and in a stronger regime the C2,τC^{2,\tau}-distance of its boundary from a sphere, is controlled linearly by the Lipschitz seminorm of the nonlocal mean curvature. Their results were stated for the singular kernel ∣z∣−n−2s|z|^{-n-2s}, with an extension to radial kernels k(∣z∣)k(|z|) under decay and monotonicity bounds, except that the linear C2,τC^{2,\tau} stability required k(t)k(t) to be a smooth perturbation of t−(n+2s)t^{-(n+2s)} at the origin; and the estimates degenerate in the local limit s→12−s\to\tfrac12^-, where the nonlocal mean curvature blows up like (1−2s)−1(1-2s)^{-1} and the analogous local stability statement is false due to bubbling. This record poses the two complementary questions that delimit the validity of that theory: determine the minimal assumptions on a general radial kernel under which the quantitative stability, in particular the linear C2,τC^{2,\tau} estimate, continues to hold, and characterize which kernel irregularities destroy it; and describe the asymptotic transition of the rigidity and stability statements as s→12−s\to\tfrac12^-, identifying whether any nontrivial uniform-in-ss stability persists across the nonlocal-to-local transition. The qualitative rigidity part for general kernels is settled, including a necessary and sufficient condition on the kernel, but the quantitative stability question for general kernels and the local-limit transition of the stability theory remain open.

Background

For a bounded open set Ω⊂Rn\Omega\subset\mathbb{R}^n and a fractional parameter s∈(0,12)s\in(0,\tfrac12), the nonlocal mean curvature of Ω\Omega at a boundary point p∈∂Ωp\in\partial\Omega is defined as HsΩ(p)=1ωn−2 PV∫Rnχ~Ω(x)∣x−p∣n+2s dx,χ~Ω=χΩc−χΩ,H_s^\Omega(p)=\frac{1}{\omega_{n-2}}\,\mathrm{PV}\int_{\mathbb{R}^n}\frac{\tilde\chi_\Omega(x)}{|x-p|^{n+2s}}\,dx,\qquad\tilde\chi_\Omega=\chi_{\Omega^c}-\chi_\Omega, where χ\chi denotes the characteristic function, ωn−2\omega_{n-2} is the surface measure of the unit (n−2)(n-2)-sphere, and the principal value PV\mathrm{PV} is defined by removing a small ball around pp and passing to the limit. It is the Euler-Lagrange (first-variation) curvature of the nonlocal perimeter Ps(Ω)=∫Ω∫Ωc∣x−y∣−n−2s dx dyP_s(\Omega)=\int_\Omega\int_{\Omega^c}|x-y|^{-n-2s}\,dx\,dy, the nonlocal analogue of the classical perimeter that appears in Ohta-Kawasaki-type energies for diblock copolymers.

The classical theory starts from Alexandrov's theorem: a connected closed embedded hypersurface with constant mean curvature is a sphere; disconnected unions of equal spheres also have constant mean curvature, so connectedness is essential in the local theory. Quantitatively, the local theory is delicate: by a gluing construction of Butscher there exist connected boundaries whose classical mean curvature is arbitrarily close to a constant in C1C^1, yet which are Hausdorff-close to unions of tangent spheres of equal radii (a phenomenon called bubbling), so no unrestricted quantitative stability holds locally; Ciraolo and Maggi obtained local quantitative stability only under additional geometric constraints such as an upper volume-density bound.

Ciraolo, Figalli, Maggi and Novaga established the nonlocal counterpart of this circle of ideas (the same rigidity was proved independently, via the method of moving planes — the reflection-symmetry method for proving symmetry of sets and solutions — by Cabré, Fall, Solà-Morales and Weth). Their Theorem 1.1: if Ω\Omega is a bounded open set of class C1,2sC^{1,2s} (the boundary is locally a C1C^1 graph whose gradient is 2s2s-Hölder continuous) and HsΩH_s^\Omega is constant on ∂Ω\partial\Omega, then ∂Ω\partial\Omega is a sphere, with no connectedness assumption, the nonlocality of the perimeter forbidding multi-bubble configurations. Their Theorem 1.2 is a sharp quantitative stability estimate: writing δs(Ω)=sup⁡p≠q∈∂Ω∣HsΩ(p)−HsΩ(q)∣∣p−q∣\delta_s(\Omega)=\sup_{p\neq q\in\partial\Omega}\frac{|H_s^\Omega(p)-H_s^\Omega(q)|}{|p-q|} for the Lipschitz-type seminorm of HsΩH_s^\Omega along ∂Ω\partial\Omega, ηs(Ω)=diam(Ω)2n+2s+1δs(Ω)/∣Ω∣2\eta_s(\Omega)=\mathrm{diam}(\Omega)^{2n+2s+1}\delta_s(\Omega)/|\Omega|^2 for the associated scale-invariant quantity, and ρ(Ω)=inf⁡{t−rdiam(Ω):  p∈Ω,  Br(p)⊂Ω⊂Bt(p)}\rho(\Omega)=\inf\Big\{\frac{t-r}{\mathrm{diam}(\Omega)}:\;p\in\Omega,\;B_r(p)\subset\Omega\subset B_t(p)\Big\} for the scale-invariant distance of Ω\Omega from a ball, one has ρ(Ω)≤C^(n) ηs(Ω)\rho(\Omega)\le\hat C(n)\,\eta_s(\Omega) whenever ∂Ω\partial\Omega is of class C2,αC^{2,\alpha} for some α>2s\alpha>2s, the exponent of ηs\eta_s being optimal (as shown by ellipsoids flattening to the unit ball). Their Theorem 1.5 upgrades this to a linear stability estimate in a strong norm: if ∂Ω\partial\Omega is of class C2,αC^{2,\alpha} for some α>2s\alpha>2s and Ω\Omega has been translated and rescaled so that B1−2ρ(Ω)(0)⊂Ω⊂B1(0)B_{1-2\rho(\Omega)}(0)\subset\Omega\subset B_1(0), then there exists a threshold η(n,s)>0\eta(n,s)>0 such that, whenever ηs(Ω)≤η(n,s)\eta_s(\Omega)\le\eta(n,s), there is a map F:∂B1(0)→RnF:\partial B_1(0)\to\mathbb{R}^n of class C2,τC^{2,\tau} — twice continuously differentiable with τ\tau-Hölder continuous second derivatives — for every τ<2s\tau<2s, with F(∂B1(0))=∂ΩF(\partial B_1(0))=\partial\Omega and ∥F−Id∥C2,τ(∂B1(0))≤C(n,s,τ) ηs(Ω)\|F-\mathrm{Id}\|_{C^{2,\tau}(\partial B_1(0))}\le C(n,s,\tau)\,\eta_s(\Omega); in particular Ω\Omega is convex. The proof linearizes the nonlocal mean curvature around the ball, showing that the linearized operator is a C1,τC^{1,\tau}-perturbation of a fractional Laplacian (the nonlocal elliptic operator (−Δ)σ(-\Delta)^\sigma, the generator of symmetric stable Lévy processes) on the sphere, and then applies Schauder-type regularity theory (a priori estimates that lift the Hölder regularity of solutions of elliptic equations) for nonlocal equations.

Two limitations are recorded in the paper itself. First, the authors note that Theorems 1.1 and 1.2 extend (with different constants and possibly without scale invariance) to nonlocal mean curvatures built from a general radial kernel k(∣z∣)k(|z|) in place of ∣z∣−n−2s|z|^{-n-2s}, provided k(t)>0k(t)>0, tn+2sk(t)+tn+2s+1∣k′(t)∣≤Ct^{n+2s}k(t)+t^{n+2s+1}|k'(t)|\le C, and sup⁡(0,t)k′≤c(t)<0\sup_{(0,t)}k'\le c(t)<0 for all t>0t>0; but the C2,τC^{2,\tau} linear stability of Theorem 1.5 needed the additional constraint that k(t)k(t) behave as a smooth perturbation of t−(n+2s)t^{-(n+2s)} as t→0+t\to0^+, that is, k(t)=t−(n+2s)g(t)k(t)=t^{-(n+2s)}g(t) for a function gg smooth in a neighbourhood of t=0t=0 with g(0)>0g(0)>0. Second, since (1−2s)HsΩ→HΩ(1-2s)H_s^\Omega\to H^\Omega, the classical mean curvature, as s→12−s\to\tfrac12^- (Abatangelo and Valdinoci), the nonlocal curvature itself blows up like (1−2s)−1(1-2s)^{-1} and the estimates degenerate, carrying no information in the local limit; and the local analog of the stability statements is false, by the bubbling counterexamples recalled above.

The subsequent literature has clarified part of this picture. On kernels: the qualitative rigidity theorem has been extended far beyond the above class — by Biswas, Foss and Radu for integrable (non-singular) kernels, where a bounded open set of class C1,βC^{1,\beta} with constant nonlocal curvature is a single ball for strictly decreasing kernels with full support, and a union of balls at mutual distance at least the support radius for compactly supported kernels; by Bucur and Fragalà for general radial non-increasing Lloc1L^1_{\mathrm{loc}} kernels under an improved-integrability condition, concluding finite unions of equal balls in general and a single ball for bounded positive decreasing kernels; and by Cygan and Grzywny, who for radial monotone kernels and C1+βC^{1+\beta} boundaries gave a necessary and sufficient condition for the Alexandrov rigidity to hold — the kernel's value at the origin must dominate its value at every positive radius, lim⁡r↓0j(r)>j(ρ)\lim_{r\downarrow0}j(r)>j(\rho) for all ρ>0\rho>0 — with explicit counterexamples when it fails; Cygan, Grzywny and Lenczewska developed a general asymptotic framework for nonlocal curvatures which, as a special case, recovers the convergence of fractional curvatures to the classical curvature as the order tends to one. On the local limit: Cesaroni, De Luca, Novaga and Ponsiglione characterized the limits of fractional mean curvature flows as s→0+s\to0^+ and s→1−s\to1^- (in the convention of the flow literature the fractional order is the parameter s∈(0,1)s\in(0,1), so that here s→1−s\to1^- is the local limit, equivalently s→12−s\to\tfrac12^- in the order-2s2s normalization used above; the convergence is level-set convergence, in the viscosity-solution formulation of geometric flows, to the classical mean curvature flow), and De Luca, Kubin and Ponsiglione treated kernels with critical and supercritical exponents via core-radius regularization, proving Gamma-convergence (the variational notion of convergence of functionals that guarantees convergence of minimizers) of the regularized nonlocal perimeters to the classical perimeter with convergence of the associated curvatures. The only quantitative Alexandrov-type estimate beyond the exact setting of the source paper, by De Gennaro, Kubin and Kubin, concerns normal deformations of a ball for the standard fractional kernel. No later work is known to revisit the kernel hypotheses of Theorem 1.5 or the behavior of the stability constants as s→12−s\to\tfrac12^-.

Problem Statement

The common objective is to delimit the domain of validity of the sharp rigidity and stability theory of Ciraolo, Figalli, Maggi and Novaga for hypersurfaces with constant or almost-constant nonlocal mean curvature, in the two directions in which that theory is currently only conditional.

Question 1 (kernel generality). For a radial kernel kk defining a nonlocal mean curvature HkΩ(p)=1ωn−2 PV∫Rn(χΩc(x)−χΩ(x)) k(∣x−p∣) dx,H_k^\Omega(p)=\frac{1}{\omega_{n-2}}\,\mathrm{PV}\int_{\mathbb{R}^n}\big(\chi_{\Omega^c}(x)-\chi_\Omega(x)\big)\,k(|x-p|)\,dx, with the almost-constancy quantity η(Ω)=diam(Ω)α δ(Ω)/∣Ω∣2\eta(\Omega)=\mathrm{diam}(\Omega)^{\alpha}\,\delta(\Omega)/|\Omega|^2 built from the Lipschitz-type seminorm δ(Ω)=sup⁡p≠q∈∂Ω∣HkΩ(p)−HkΩ(q)∣/∣p−q∣\delta(\Omega)=\sup_{p\neq q\in\partial\Omega}|H_k^\Omega(p)-H_k^\Omega(q)|/|p-q| of HkΩH_k^\Omega along ∂Ω\partial\Omega, where the exponent α\alpha is prescribed by the homogeneity class assumed of kk so that η(Ω)\eta(\Omega) is scale-invariant (for the model kernel k(t)=t−(n+2s)k(t)=t^{-(n+2s)} this gives α=2n+2s+1\alpha=2n+2s+1, recovering ηs(Ω)\eta_s(\Omega)), determine the minimal structural and regularity assumptions on kk under which the quantitative stability conclusions persist: the closeness of Ω\Omega to a single ball (Theorem 1.2 type) and, above all, the linear C2,τC^{2,\tau} stability (Theorem 1.5 type), ∥F−Id∥C2,τ≤C η(Ω)\|F-\mathrm{Id}\|_{C^{2,\tau}}\le C\,\eta(\Omega) for η(Ω)\eta(\Omega) small. Within this, characterize precisely which kernel singularities or irregularities destroy the linear higher-regularity estimate: which hypotheses on kk are genuinely necessary, and for which kernels does qualitative rigidity hold while the quantitative C2,τC^{2,\tau} stability fails. Since the 1.5-type linear C2,τC^{2,\tau} estimate implies the 1.2-type closeness to a single ball, a resolution of Question 1 consists of that estimate for the claimed class of kernels together with the necessity claim on the hypotheses on kk.

Question 2 (the local limit). Describe the asymptotic transition as s→12−s\to\tfrac12^- of the nonlocal rigidity and stability statements above: how the constants C(n,s,τ)C(n,s,\tau), the smallness thresholds η(n,s)\eta(n,s), and the scale-invariance properties degenerate as the order tends to the local regime, in which the curvature rescales by (1−2s)−1(1-2s)^{-1} and the limiting local statement is false by the bubbling counterexamples; and determine whether any nontrivial uniform-in-ss stability phenomenon persists across the nonlocal-to-local transition. A resolution of Question 2 consists of the asymptotics of the constants and thresholds together with a determination of whether a renormalized uniform-in-ss estimate holds.

The two questions are independent in method but jointly delimit the same theory; a resolution of the record requires addressing both.

References

  1. Giulio Ciraolo, Alessio Figalli, Francesco Maggi, Matteo Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, Journal für die reine und angewandte Mathematik (Crelles Journal) 741 (2018), 275-294. DOI: 10.1515/crelle-2015-0088, https://doi.org/10.1515/crelle-2015-0088 (arXiv:1503.00653, https://arxiv.org/abs/1503.00653).
  2. Xavier Cabré, Mouhamed Moustapha Fall, Joan Solà-Morales, Tobias Weth, Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay, Journal für die reine und angewandte Mathematik (Crelles Journal) 745 (2018), 253-280. DOI: 10.1515/crelle-2015-0117, https://doi.org/10.1515/crelle-2015-0117 (arXiv:1503.00469, https://arxiv.org/abs/1503.00469).
  3. Nicola Abatangelo, Enrico Valdinoci, A Notion of Nonlocal Curvature, Numerical Functional Analysis and Optimization 35 (2014), no. 7-9, 793-815. DOI: 10.1080/01630563.2014.901837, https://doi.org/10.1080/01630563.2014.901837.
  4. Adrian Butscher, A gluing construction for prescribed mean curvature, Pacific Journal of Mathematics 249 (2011), 257-269. DOI: 10.2140/pjm.2011.249.257, https://doi.org/10.2140/pjm.2011.249.257.
  5. Giulio Ciraolo, Francesco Maggi, On the Shape of Compact Hypersurfaces with Almost-Constant Mean Curvature, Communications on Pure and Applied Mathematics 70 (2017), no. 4, 665-716. DOI: 10.1002/cpa.21683, https://doi.org/10.1002/cpa.21683.
  6. Animesh Biswas, Mikil Foss, Petronela Radu, Nonlocal Mean Curvature with Integrable Kernel (2022). arXiv:2212.01530, https://arxiv.org/abs/2212.01530.
  7. Animesh Biswas, Mikil Foss, Petronela Radu, Nonlocal Ordered Mean Curvature with Integrable Kernel (2025). arXiv:2501.16623, https://arxiv.org/abs/2501.16623.
  8. Dorin Bucur, Ilaria Fragalà, Alexandrov theorem for general nonlocal curvatures: the geometric impact of the kernel, Journal de Mathématiques Pures et Appliquées 169 (2023), 82-108. DOI: 10.1016/j.matpur.2022.11.008, https://doi.org/10.1016/j.matpur.2022.11.008 (arXiv:2202.03180, https://arxiv.org/abs/2202.03180).
  9. Wojciech Cygan, Tomasz Grzywny, Alexandrov Theorem for nonlocal curvature (2024). arXiv:2410.10199, https://arxiv.org/abs/2410.10199.
  10. Wojciech Cygan, Tomasz Grzywny, Julia Lenczewska, Asymptotics and geometric flows for a class of nonlocal curvatures, Fractional Calculus and Applied Analysis 29 (2026), no. 1, 494-542. DOI: 10.1007/s13540-026-00486-w, https://doi.org/10.1007/s13540-026-00486-w (arXiv:2308.16660, https://arxiv.org/abs/2308.16660).
  11. Annalisa Cesaroni, Lucia De Luca, Matteo Novaga, Marcello Ponsiglione, Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (2020). arXiv:2003.02248, https://arxiv.org/abs/2003.02248.
  12. Lucia De Luca, Andrea Kubin, Marcello Ponsiglione, Convergence of supercritical fractional flows to the mean curvature flow (2021). arXiv:2102.13041, https://arxiv.org/abs/2102.13041.
  13. Daniele De Gennaro, Andrea Kubin, Anna Kubin, Asymptotic of the discrete volume-preserving fractional mean curvature flow via a nonlocal quantitative Alexandrov theorem, Nonlinear Analysis 228 (2023), 113200. DOI: 10.1016/j.na.2022.113200, https://doi.org/10.1016/j.na.2022.113200 (arXiv:2204.07450, https://arxiv.org/abs/2204.07450).
  14. Francesc Alcover, Renzo Bruera, Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature (2026). arXiv:2602.18219, https://arxiv.org/abs/2602.18219.
  15. Jack Thompson, Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates, PhD thesis (2025). arXiv:2507.09219, https://arxiv.org/abs/2507.09219 (summary published in Bulletin of the Australian Mathematical Society, DOI: 10.1017/S0004972725100270, https://doi.org/10.1017/S0004972725100270).