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Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Under the anisotropic rescaling of the Heisenberg group, a nonlocal mean-field equation has zero level sets that evolve by horizontal mean curvature flow $V_0=\theta\kappa_0$.

desk verdict A genuinely new formal derivation of horizontal MCF from a nonlocal mean-field equation in H1 and SE(2), but the printed θ formula is off by a factor 1/2 and Theorem 4.1 is only a conditional formal statement. read the letter →

arxiv 2411.15814 v1 pith:S6NZNEHH submitted 2024-11-24 math.AP

Giovanna Citti , Nicolas Dirr , Federica Dragoni , Raffaele Grande This is my paper · ORCID
classification math.AP MSC 35K9335R0353C1735B25
keywords meancurvatureflowHeisenberggroupnonlocalmean-fieldequationasymptoticexpansionsub-RiemanniangeometryCarnotgroupsroto-translationvisualcortexmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that, under the anisotropic dilations of the Heisenberg group, solutions of the nonlocal mean-field equation $\varepsilon^{2}\partial_t m_\varepsilon=-m_\varepsilon+\tanh(\beta\int J^\varepsilon(y^{-1}\circ x)\,m_\varepsilon(y)\,dy)$ have zero level sets that move, away from characteristic points, by horizontal mean curvature flow $V_0=\theta\kappa_0$, with $\theta$ the explicit kernel-dependent constant in (21). The derivation is a formal asymptotic expansion: a one-dimensional instanton profile is placed along the signed distance to the moving surface, and the leading-order part of the expanded convolution produces a curvature term through the horizontal Hessian of the distance. The paper states the convergence as Theorem 4.1, conditional on a regularity and comparison assumption for $\delta$-perturbed surfaces (Definition 4.2), and stresses that this assumption is needed because characteristic points can be singular even for smooth surfaces. The same calculation extends locally to the roto-translation group $\mathrm{SE}(2)$, and numerical experiments reproduce an exact axisymmetric shrinking solution, including motion at characteristic points.

What carries the argument

Three ingredients carry the argument. First, the anisotropic dilation $\delta_\lambda(x_1,x_2,x_3)=(\lambda x_1,\lambda x_2,\lambda^2 x_3)$ and the rescaled kernel $J^\varepsilon(x)=\varepsilon^{-4}J(\delta_{1/\varepsilon}x)$, with homogeneous dimension $Q=4$, make the nonlocal convolution compatible with the sub-Riemannian geometry. Second, the one-dimensional instanton $m$, the odd monotone solution of $-m+\tanh(\beta J*m)=0$ on the line, whose derivative $m'$ is the zero eigenfunction of the linearized operator $Lf=-f+(1-m^2)\int J(s)f(r+s)\,ds$; solvability of the first corrector is controlled by the $L^2(\mu)$-orthogonality condition of Lemma 2.3. Third, the homogeneous Taylor expansion of the signed distance, whose symmetrized horizontal Hessian has eigenvalue $0$ along the horizontal normal and eigenvalue $\kappa_0$ along the orthogonal horizontal direction, so the surviving quadratic term in the expanded convolution is exactly the curvature term. Projecting the curvature remainder onto $m'$ fixes the speed coefficient $\theta$ in (21); the corresponding argument on the Lie algebra with local dilations gives the $\mathrm{SE}(2)$ extension.

What would settle it

Take a compact, non-axisymmetric initial surface in $\mathbb{H}^1$ with a known non-characteristic point, solve (15) numerically for decreasing $\varepsilon$, and measure the normal velocity of the zero level set at that point; the claim predicts $V_0=\theta\kappa_0$ with $\theta$ from (21). A persistent deviation beyond discretization error as $\varepsilon\to0$ would falsify the scaling limit. A complementary check is whether any family $M_t^\delta$ satisfying Definition 4.2 can exist up to the claimed time: if a surface develops a characteristic singularity before time $T$, the theorem's main hypothesis fails, and the formal expansion supplies no error estimate to replace it.

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Extended reading notes

Core claim

The central claim is that the anisotropic scaling limit of a nonlocal mean-field equation provides a new regularisation and approximation of Heisenberg mean curvature flow. Concretely, if $m_\varepsilon$ solves (15) with kernel $J^\varepsilon(x)=\varepsilon^{-4}J(\delta_{1/\varepsilon}x)$, then as $\varepsilon\to0$ the zero level set of $m_\varepsilon$ tends, locally uniformly outside $\partial M_t$, to the surface $\partial M_t$ evolving by $V_0=\theta\kappa_0$, where $\kappa_0=\operatorname{div}_{\mathbb{H}^1} n_0$ is the horizontal mean curvature and $\theta$ is the $L^2(\mu)$-projection constant in (21). The theorem is conditional: it assumes the existence of smooth surfaces $M_t^\delta$ without characteristic points that satisfy $V_0=\theta\kappa_0+\delta$ classically and obey a comparison principle, with Hausdorff distance to $M_t$ vanishing as $\delta\to0$. The paper presents the expansion as a formal argument at non-characteristic points and offers numerical evidence that the approximation persists near characteristic points, where the flow itself is not classically defined. By freezing the Lie algebra, the same expansion is transferred to the roto-translation group $\mathrm{SE}(2)$.

Load-bearing premise

The load-bearing premise is that, for small $\delta$, the evolving surface can be surrounded by smooth surfaces without characteristic points that move with speed $\theta\kappa_0+\delta$, obey a comparison principle, and converge to the true surface as $\delta\to0$; the paper explicitly labels this an assumption because characteristic points can be singular even for smooth surfaces.

Editorial extensions

If this is right

  • For small but finite $\varepsilon$, the nonlocal equation is globally defined and gives a regularisation of Heisenberg mean curvature flow rooted in a multiscale derivation.
  • Away from characteristic points, the effective speed is $V_0=\theta\kappa_0$, so the coefficient $\theta$ from (21) determines the time scale of the limiting flow for a given kernel and inverse temperature.
  • The same derivation applies to the roto-translation group $\mathrm{SE}(2)$, so the nonlocal equation approximates the sub-Riemannian curvature flow used in a cortical model of image completion and denoising.
  • The numerical scheme interpolates between a nonlocal mean-field evolution and a diffusion-concentration thresholding algorithm; in the appropriate parameter limits it reduces to those two classical procedures.
  • The exact axisymmetric shrinking solution with characteristic points is reproduced numerically, indicating that the approximation is meaningful even where the curvature flow is classically undefined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct consequence the authors leave implicit is that the numerical scheme can be tuned through the kernel and inverse temperature to produce a prescribed speed coefficient $\theta$; measuring $\theta$ from (21) for a compactly supported kernel would be a cheap experimental check.
  • The argument suggests a route to a fully rigorous convergence theorem if a comparison principle for Heisenberg mean curvature flow were available in the needed generality; the paper identifies exactly this gap when it labels Definition 4.2 an assumption.
  • The same formal machinery should apply to other step-two Carnot or sub-Riemannian structures by replacing the Heisenberg dilation with the appropriate homogeneous dilation, with characteristic points again the main obstruction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper derives a formal asymptotic expansion for the nonlocal mean-field equation (15) under the anisotropic Heisenberg rescaling and concludes that, away from characteristic points, the zero level set evolves by horizontal mean curvature flow with normal velocity V0 = θ k0, where θ is given by (21). The same formal derivation is transferred to SE(2) by freezing the Lie algebra. Numerical experiments with the Heisenberg heat kernel are reported, including a test against an exact axisymmetric solution with characteristic points from [22]. The paper explicitly presents the result as a regularization of Heisenberg mean curvature flow and does not claim a full rigorous convergence proof.

Significance. If the derivation and coefficient were correct, the paper would give a new multiscale link from a particle/mean-field model to sub-Riemannian curvature flow, with potential relevance to cortical models of V1. The paper also proposes a numerical scheme interpolating between an Ermentrout-Cowan type equation and a Bence-Merriman-Osher type algorithm, and it reproduces known exact solutions. These are valuable contributions. However, two load-bearing issues affect the central claim: the printed formula for θ appears to be off by a factor of 1/2 relative to the expansion in §4.1, and Theorem 4.1 asserts convergence although only a formal expansion is provided and the Euclidean trapping argument is not transferred to the Heisenberg setting.

major comments (2)
  1. [§4.1, Eq. (21)] Formula (21) is inconsistent with the expansion preceding it. In (20) the second-order term carries a factor ε/2, and immediately afterwards R(r) is defined as R(r) = (1/2) ∫ Ĵ(ŷ) m′(r+ŷ1) ŷ2² dŷ. Applying the solvability condition from Lemma 2.3, namely ⟨(1−m²)k0 R + L(m1), m′/(1−m²)⟩ = 0, gives V0 = k0 θ with θ = (β/(2N)) ∫ m′(r)m′(r+r1) Ĵ(r1²+s²) s² dr dr1 ds. The factor 1/2 is missing in the printed expression (21). The numerical validation in §4.2 determines θ by linear regression on a shrinking cylinder and never evaluates (21), so the numerics cannot detect the factor error. This is a load-bearing issue because Theorem 4.1 refers to 'θ as in (21)', and the stated interface speed would be twice the speed derived from the paper's own expansion.
  2. [Theorem 4.1 and Definition 4.2] Theorem 4.1 states a locally uniform convergence result, but the proof supplied in §4.1 is only a formal asymptotic expansion with no error estimates. The Euclidean proof summarized in Section 2 relies on comparison principles and a trapping argument for biased solutions; in the Heisenberg setting the paper itself notes (Section 3 and after Definition 4.2) that comparison principles for horizontal mean curvature flow are only partially known. Definition 4.2 assumes the existence of smooth sets M_t^δ satisfying V0 = θκ0 + δ and a comparison principle, but this assumption does not by itself imply convergence of the solutions mε of (15) to the indicator of M_t. No argument is given that the zero level sets of mε are trapped between M_t^δ and M_t^{-δ}. Thus the theorem, as stated, is not proven by the material in §4.1; it should either be downgraded to a formal/conjectural statement or supplied with the missing convergence argument.
minor comments (5)
  1. [Theorem 4.1] The convergence statement says 'locally uniformly in RN \ ∂Mt', but the setting is R3; the formula should read R3 (or H1), and the two limit values should be stated consistently with the conventions for Mt and its complement.
  2. [Definition 4.2] The phrase 'M_0^δ has Hausdorff distance δ from M0' is imprecise: Hausdorff distance is between sets, and it is not clear whether the intended meaning is dist_H(M0^δ, M0) = δ or dist_H(∂M0^δ, ∂M0) = δ. Please clarify.
  3. [Abstract and §4.2] There are typographical errors: 'Ementrout-Cowan' in the abstract and 'mena curvature flow' in §4.2 should be 'Ermentrout-Cowan' and 'mean curvature flow', respectively.
  4. [References] Reference [25] is cited as 'Katzoulakis' in the bibliography; the correct spelling is 'Katsoulakis'.
  5. [§4.1 notation] The notation alternates between R(r) and 'hat R' in the splitting step before equation (21); please use a single symbol consistently, especially since the missing factor 1/2 is tied to the definition of R.

Circularity Check

1 steps flagged · score 4.0 of 10

The formal expansion is non-circular, but the numerical validation fits θ from a cylinder and then uses that fitted value in the exact sphere solution, so the reported check does not independently test the first-principles formula (21); an apparent 1/2 factor inconsistency in (21) is masked by this calibration.

  1. fitted input called prediction [Section 4.2, numerical validation (paragraphs before Figures 1–2)]
    "The next step is to determine the factor θ. A cylinder revolving around the x3-axis evolves according to the 2-dimensional Euclidean mean curvature flow. ... we find θ = 0.56561 by a linear regression. Then we compute the nonlocal evolution equation with the heat kernel and initial condition m((∥x∥H1 − r)/ε), which has ∂Br as its zero level set. The exact solution at time t is obtained as (see [22]) ∂Mt = {x ∈ H1 : (x21+x22)2 + 12θt(x21+x22) + 16x23 + 12(θt)2 = r4}."

    The coefficient θ is not evaluated from the derived expression (21); it is fitted by linear regression to a cylinder. The same fitted value is then inserted into the exact sphere solution from [22] and compared with the level set of the nonlocal equation. The sphere test is therefore a one-parameter consistency check of the nonlocal equation against Heisenberg mean curvature flow with an already fitted mobility, not a prediction of θ from first principles. In particular it cannot detect an error in (21): comparing (20) with R(r)=1/2∫ Ĵ m′ ŷ2² and the solvability condition VN=k0∫m′R gives a factor 1/2 missing from the printed (21), and the calibration would absorb any such factor.

full rationale

The asymptotic derivation in §4.1 is not circular by construction: it uses the external instanton and spectral-gap lemmas from [15,32], expands the Heisenberg convolution (17)–(20), and obtains the interface speed from the solvability condition for L(m1). Definition 4.2 is explicitly labeled an assumption, so Theorem 4.1's conditionality is not a disguised use of the conclusion. The main circularity-adjacent step is numerical: §4.2 determines θ=0.56561 by fitting a cylinder evolution and then uses that fitted θ in the exact sphere solution from [22]; the sphere plot is a consistency check with a calibrated parameter, not an independent test of formula (21). The printed formula (21) also appears to omit the factor 1/2 present in R(r), which strengthens the concern because the numerics never evaluate (21). Self-citations ([3,8,13,14,18,23]) are contextual and not load-bearing for the derivation; the explicit sphere solution and its uniqueness come from external [22]. Overall score 4: partial circularity in the numerical evidence, while the formal expansion retains independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the instanton and spectral-gap lemmas imported from [15,32], on the distance Hessian structure from [2], and on a kernel symmetry assumption. The paper adds a strong, unproved regularity and comparison assumption on the limit evolution, and the numerical section introduces fitted values for theta and for the instanton profile rather than evaluating the theoretical formulas.

free parameters (3)
  • theta (effective horizontal surface tension) = 0.56561 (linear regression, Section 4.2)
    In the numerical validation theta is determined by fitting a cylinder's radius decay to r'(t) = -theta/r rather than evaluating formula (21); the sphere comparison then uses this fitted value.
  • inverse temperature beta = 1.2
    Chosen for the numerics; fixes m_beta = 0.6585. Not fitted to the target solution, but a hand-chosen model parameter.
  • instanton profile m = computed by stabilization of a planar front (Section 4.2)
    Used as the initial profile and in the ansatz; obtained numerically instead of from a closed-form expression, although existence and properties are established in Lemma 2.2.
assumptions (5)
  • standard math Existence, uniqueness, symmetry and exponential decay of the 1D instanton m solving -m + tanh(beta J * m) = 0 (Lemma 2.2).
    Taken from [15,32]; used to build the ansatz in Sections 2 and 4.1.
  • standard math Spectral gap and solvability condition for the linearized operator L in L2(mu) (Lemma 2.3).
    From [15]; used to solve for the correction m1 and to define theta via the projection on m'.
  • standard math Hessian structure of the signed distance in H1: zero eigenvalue in the normal direction and trace equal to horizontal mean curvature at non-characteristic points (Theorem 3.1 of [2]).
    Used in the convolution expansion (19) to reduce the Hessian term to k0(x0).
  • domain assumption Kernel symmetry: J(x1,x2,x3) = J(x1^2+x2^2, |x3|) (Definition 4.1).
    Needed so the reduced kernel is radially symmetric and the Hessian term reduces to curvature; without it anisotropic terms would appear.
  • ad hoc to paper Existence of a classical evolution M_t^delta with comparison principle satisfying V0 = theta kappa0 + delta (Definition 4.2).
    This is assumed, not proved; the paper states it is an assumption because characteristic points may act as singularities even for smooth surfaces.

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Pith. "Pith review of Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group." pith.science (2026). https://pith.science/paper/S6NZNEHH

@misc{pith2026241115814,
  author       = {Pith},
  title        = {Pith review of: Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6NZNEHH}},
  note         = {Machine review of arXiv:2411.15814}
}
read the original abstract

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multiscale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the models of the visual cortex proposed by Petitot and Citti-Sarti, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field equation to curvature flows which are used in image processing. From a pure mathematical point of view, it provides a new approximation and regularization of Heisenberg mean curvature flow. Using the local structure of the rototranslational group, we extend the result to cover the model by Citti and Sarti. Numerically, the parameters in our algorithm interpolate between solving an Ementrout-Cowan type of equation and a Bence-Merriman-Osher algorithm type algorithm for sub-Riemannian mean curvature. We also reproduce some known exact solutions in the Heisenberg case.

Figures

Figures reproduced from arXiv: 2411.15814 by the authors.

Figure 1
Figure 1. Computation of θ, x-axis time, y-axis radius (left) and area (right) of circle [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Zero level sets of mε (t, x1, 0, x3) for different times, both for exact solution of the Heisenberg MCF (black) and the level set of the numerically solved nonlocal evolution equation (blue). In [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Graph of mε (t, x1, 0, 0) (solid) and mε (t, 0, 0, x3) (dashed) at t = 0.32 5 Extension to the SE(2) group In this section we show how to extend the result to the group SE(2). Since the Heisenberg group is the tangent cone to the roto-translation group, they present the same local approximation. Here we briefly introduce the group and refer to [14] for more results. We will identify the group with the space R 2 × S … view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [22]

    Ferrari, Q

    F. Ferrari, Q. Liu and J. Manfredi On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group . Communications in Contemporary Mathematics, Vol. 16, Issue 3, 2014

    work page 2014

  2. [1]

    Alikakos, P

    N. Alikakos, P. Bates and X. Chen. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model . Archive for Rational Mechanics and Analysis, vol. 128, pp. 165–205, 1994

    work page 1994

  3. [2]

    Arcozzi, F.Ferrari

    N. Arcozzi, F.Ferrari. The Hessian of the distance from a surface in the Heisenberg group. Annales Academiæ Scientiarum Fennicæ Mathematica, Vol 33, pp. 35–63, 2008

    work page 2008

  4. [3]

    Baspinar and G

    E. Baspinar and G. Citti. Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures . SIAM Journal on Mathemat- ical Analysis, Vol 51, Issue 3, 2019

    work page 2019

  5. [4]

    P. C. Bressloff, J. D. Cowan, The functional geometry of local and long- range connections in a model of V1. J. Physiol. Paris, 97, 2-3, 221-236, 2003. 21

    work page 2003

  6. [5]

    Asymptotics for the heat kernel on H-type groups Annali di Matematica Pura ed Applicata, volume 197, pp

    T.Bruno and M.Calzi. Asymptotics for the heat kernel on H-type groups Annali di Matematica Pura ed Applicata, volume 197, pp. 1017–1049, 2018

    work page 2018

  7. [6]

    Bonfiglioli

    A. Bonfiglioli. Taylor formula for homogenous groups and applications Mathematische Zeitschrift, vo. 262, 2009

    work page 2009

  8. [7]

    Bonfiglioli, E

    A. Bonfiglioli, E. Lanconelli and F. Uguzzoni. Stratified Lie Group and Potential Theory for their Sub-Laplacian. Springer, 2007

    work page 2007

Show all 33 references
  1. [8]

    Capogna and G

    L. Capogna and G. Citti. Generalized mean curvature flow in Carnot groups. Comm. Partial Differential Equations, vol. 34, Issue 8, pp. 937-

  2. [9]

    Carlen, M.C

    E. Carlen, M.C. Carvalho, E. Orlandi. Approximate Solutions of the Cahn- Hilliard Equation via Corrections to the Mullins-Sekerka Motion Archive for Rational Mechanics and Analysis, vol 178 pp 1-55, 2005

    2005

  3. [10]

    Free Energy of a Nonuniform System

    J.Cahn, J.Hilliard. Free Energy of a Nonuniform System. I. Interfacial Free Energy. The Journal of Chemical Physics. AIP Publishing. vol 28, pp. 258–267, 1958

    1958

  4. [11]

    Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

    Y.Chen, Y.Giga, S.Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geometry, vol 33, Issue 3, pp. 749-786, 1991

    1991

  5. [12]

    W.L. Chow. ¨Uber systeme von linearen partiellen differentialgleichungen erster ordnung . Math Ann. vol 117, 1939

    1939

  6. [13]

    Citti, B

    G. Citti, B. Franceschiello, G. Sanguinetti and A. Sarti. Sub-Riemannian mean curvature flow for image processing . SIAM Journal Imaging Sci., vol 9, Issue 1, pp. 212–237, 2016

    2016

  7. [14]

    Citti and A

    G. Citti and A. Sarti. A Cortical Based Model of Perceptual Completion in the Roto-Translation Space. Journal of Mathematical Imaging and Vision, vol 24, pp. 307–326, 2006

    2006

  8. [15]

    De Masi, E

    A. De Masi, E. Orlandi, E. Presutti, L. Triolo, Motion by curvature by scaling nonlocal evolution equations . Journal of Statistical Physics, vol 73, pp. 543–570, 1993

    1993

  9. [16]

    De Mottoni, M

    P. De Mottoni, M. Schatzman. Geometrical Evolution of Developed Inter- faces. Transactions of the American Mathematical Society, Vol. 347, Issue 5, pp. 1533-1589, 1995

    1995

  10. [17]

    N. Dirr. A Stefan problem with surface tension as the sharp interface limit of a nonlocal system of phase-field type. Journal of statistical physics 114 (3-4) , pp. 1085-1113, 2004

    2004

  11. [18]

    N. Dirr, F. Dragoni and M. von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries . Communications on Pure and Applied Mathematics, vol 9, Issue 2, pp. 307-326, 2010

    2010

  12. [19]

    Metric Hopf-Lax formula with semicontinuous data

    F.Dragoni. Metric Hopf-Lax formula with semicontinuous data . Discrete and Continuous Dynamical Systems, vol 17, Issue 4, 2007 22

    2007

  13. [20]

    Evans and J

    L.C. Evans and J. Spruck. Motion of level sets by mean curvature . Journal of Differential Geometry, vol 33, Issue 3 pp.635-681, 1991

    1991

  14. [21]

    G. B. Ermentrout and J. D. Cowan. A mathematical theory of visual hal- lucination patterns. Biological Cybernetics, vol 34, pp. 137–150, 1979

    1979

  15. [23]

    R. Grande. A stochastic representation for the solution of approximated mean curvature flow . Nonlinear Differential Equations and Applications NoDEA, vol 29, Issue 1, 2022

    2022

  16. [24]

    M. A. Katsoulakis and A. T. Kho Stochastic curvature flows: asymptotic derivation, level set formulation and numerical experiments , Interfaces and Free Boundaries, vol. 3 pp. 65-290, 2001

    2001

  17. [25]

    Communications in Mathematical Physics, vol 169, vol 1, pp

    M.Katzoulakis, P.Souganidis Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics. Communications in Mathematical Physics, vol 169, vol 1, pp. 61-97, 1995

    1995

  18. [26]

    Lacoin, F

    H. Lacoin, F. Simenhaus and F. L. Toninelli. Zero-temperature 2D stochas- tic Ising model and anisotropic curve-shortening flow , Journal of the Eu- ropean Mathematical Society, vol. 16, pp. 2557-2615, 2014,

    2014

  19. [27]

    Le Donne, Lecture notes on subRiemannian geometry - Carnot- Carath´ eodory spaces from the Lie Group viewpoint

    E. Le Donne, Lecture notes on subRiemannian geometry - Carnot- Carath´ eodory spaces from the Lie Group viewpoint

  20. [28]

    Montgomery

    R. Montgomery. A Tour of sub-Riemannian Geometries, their Geodesics and Applications . Math. Surv. and Monographs 91 AMS, 2000

    2000

  21. [29]

    Serra Cassano Surface measures in Carnot-Carath` eodory spaces Calc

    R.Monti, F. Serra Cassano Surface measures in Carnot-Carath` eodory spaces Calc. Var. 13, 339–376 (2001)

    2001

  22. [30]

    J. Petitot. Elements of Neurogeometry, Functional Architectures of Vision Springer, 2017

    2017

  23. [31]

    Petitot, Y

    J. Petitot, Y. Tondut. Vers une neurog´ eom´ etrie. Fibrations corticales, structures de contact et contours subjectifs modaux. Math´ ematiques Infor- matique et Science Humaines, Vol 145 (1999), pp. 5-101

    1999

  24. [32]

    Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics

    E.Presutti. Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, 2009

    2009

  25. [33]

    L. P. Rothschild and E. M. Stein. Hypoeliptic operators and nilpotent groups, Acta Mathematica, vol. 137, (1976) , pp. 247-320. 23

    1976

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