Coarse nodal counts on sub-Riemannian manifolds

Irene Silvestre-Rosell\'{o}; Vuka\v{s}in Stojisavljevi\'{c}

Coarse nodal counts on sub-Riemannian manifolds

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Pith's one-line read This paper proves coarse, persistence-based versions of Courant's nodal domain theorem and Bézout's theorem for linear combinations of sub-Laplacian eigenfunctions on compact nilmanifolds, with exponents arbitrarily close to the conjectured

desk verdict First coarse Courant and Bézout bounds for sub-Laplacians on nilmanifolds; the proof is careful and the main external dependence is a published, uniformly stated approximation theorem, not a hidden gap. read the letter →

arxiv 2608.01446 v1 pith:ENEOXBHB submitted 2026-08-02 math.SP math.APmath.AT

Irene Silvestre-Roselló , Vukašin Stojisavljević This is my paper · ORCID

classification math.SPmath.APmath.AT MSC 35H2035P1553C1758J5055N31

keywords coarsenodalcounttopologicalpersistencesub-RiemanniangeometryCarnotgroupssub-LaplacianeigenfunctionsCourant-typetheoremsanisotropicSobolevspaces

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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 extends the coarse (persistence-based) Courant and Bézout bounds, previously known on Riemannian manifolds, to sub-Riemannian settings. On compact nilmanifolds—quotients of Carnot groups by lattices—it proves that the number of δ-deep nodal domains (and higher-degree coarse counts) of an $L^{2}$-normalized linear combination of sub-Laplacian eigenfunctions with eigenvalues ≤ λ grows at most like $δ^{{-2ε_k}}$(λ+1)^{Q/2(1+ε_k)}, where Q is the homogeneous dimension and ε_k can be made arbitrarily small by taking k large. The same method yields a coarse Bézout bound for the zero sets of such combinations, and all results extend to general maximally hypoelliptic operators. The proof combines topological persistence with anisotropic Sobolev theory of bracket-generating vector fields; a conjectured optimal bound for general equiregular sub-Riemannian manifolds is stated and a programme for proving it outlined, with the missing ingredient identified as an equiregular analogue of the polynomial approximation theorem.

What carries the argument

The principal objects are the coarse nodal count m_{d,δ}(F) = rank( Ȟ_d({|F|>δ}) → Ȟ_d({F≠0}) ) and the coarse zero count z_{d,δ}(F) = rank( Ȟ_d({F=0}) → Ȟ_d({|F|≤δ}) ), which for d=0 count δ-deep nodal domains and components of a δ-thickening that touch the zero set. These are controlled through the persistence barcode of ±|F|, whose bar-counting invariant N_{d,δ} obeys subadditivity, stability, and a duality between degree d and n−1−d. The new analytic input is the Morrey–Sobolev approximation theorem on Carnot balls (Theorem 5.3): a $C^{0}$ approximation of each f_i by polynomials of homogeneous degree k−1 with error C $r^{{k−Q/p}}$||X^k f||_{L^p}, which lets each ball in a cover be replaced

What would settle it

On the Heisenberg nilmanifold, take a fixed smooth F (e.g., the projection of sin(2πx)cos(2πy) with $L^{2}$ normalization) and compute the degree-0 bar count N_{0,δ}(|F|) as δ→0. If it grows faster than $δ^{{-Qp/(kp−Q)}}$ for some kp>Q while the anisotropic $W^{{k,p}}$_X norm stays bounded, Theorem 1.5's exponent is wrong. Similarly, a family of eigenfunction combinations with m_{0,δ} exceeding the $δ^{{-2ε_k}}$(λ+1)^{Q/2(1+ε_k)} growth would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

On a closed sub-Riemannian nilmanifold N obtained as a lattice quotient of a stratified (Carnot) group, the authors prove a quantitative bound for the coarse topology of nodal sets of vector-valued functions F=(f_1,...,f_l) whose components lie in the spectral subspace F_λ of the sub-Laplacian. The central estimate (Theorem 1.5) is an anisotropic Sobolev inequality for persistence barcodes: for any smooth map F and any k,p with kp>Q, the number of bars of length >δ in the degree-0 (and degree n−1) barcode of ±|F| is bounded by C(||F||_{$W^{{k,p}}$_X}/δ)^{Qp/(kp−Q)} plus the Betti number. From this, Theorem 1.1 gives, for every k>Q/2 and d∈{0,n−1}, the coarse Courant bound m_{d,δ}(F) ≤ C δ^{-2ε_k

Load-bearing premise

Everything rests on the Morrey–Sobolev approximation theorem (Theorem 5.3): on every Carnot ball, a smooth function can be uniformly approximated by a stratified polynomial of controlled degree with error proportional to the ball radius raised to k−Q/p; if this approximation cannot be made uniform over the covering balls, the barcode estimate and the nodal bounds collapse.

Editorial extensions

If this is right

On nilmanifolds, the coarse Courant count of δ-deep nodal domains for L^2-normalized combinations of eigenfunctions with eigenvalues ≤λ is at most C δ^{-2ε_k}(λ+1)^{Q/2(1+ε_k)}, with ε_k arbitrarily small for large k.
The same proof yields the coarse Bézout count of intersections of l nodal sets, with the same growth.
Both bounds extend from sub-Laplacians to any maximally hypoelliptic operator of order r>Q/2, with exponent 2Q/(2r−Q).
On general closed equiregular sub-Riemannian manifolds, the combination of the Riemannian result with subelliptic estimates gives a bound with exponent sn/2 instead of Q/2, confirming the need for genuinely sub-Riemannian methods.
The programme identifies three concrete gaps—an equiregular polynomial approximation, a multiscale subdivision, and covers with controlled intersections—each of which would carry the result from nilmanifolds to the general case.

Reading between the lines

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

The δ-dependence δ^{-2ε_k} weakens as k grows, suggesting that in the conjectured limit the coarse count is governed by a fixed power of δ independent of the spectral window; a sharp version would make the δ and λ factors nearly independent.
The same barcode-plus-approximation strategy could be applied to solutions of other hypoelliptic equations or to functions with merely measurable anisotropic regularity, yielding analogue bounds without spectral structure.
The duality between degree 0 and degree n−1 is a free corollary of persistence duality; extending to intermediate degrees will require controlling the topology of intersections of the covering balls, a problem of independent interest in quantitative topology.
If the equiregular polynomial approximation is ever established, the rest of the programme is largely a technical extension, meaning the conjecture's hardest part is analytic (approximation theory), not topological.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is the real thing. It proves the first coarse Courant and Bézout bounds for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds, with the expected exponent up to a factor (1+ε_k). The engine is a barcode estimate for anisotropic Sobolev norms (Theorem 1.5), and the deduction to the spectral results goes through standard subelliptic coercivity and persistence inequalities. I checked the main chain and found no gap. The paper also extends to maximally hypoelliptic operators and handles homological degree n−1 via the duality in Proposition 2.20, which is a nice bonus.

What is genuinely new: the combination of topological persistence with anisotropic Sobolev theory and local polynomial approximation on stratified groups is not a routine translation of the Riemannian argument. The Riemannian case [14] uses dyadic cube partitions; here the covering argument uses homogeneous balls and the ball-box theorem. Theorem 1.5 is the real content, and its proof is carefully written.

Soft spots, in proportion. The main external dependence is Theorem 5.3 (Lu–Wheeden Morrey–Sobolev approximation). The stress-test concern about whether the constant is uniform in the ball center is fair in principle, but the theorem is stated in exactly the uniform ball-local form and it is a published result; the paper cites [38,39] and [52] for it. I would not call this a gap, just a dependency that is not re-proved. If I were refereeing, I'd ask the authors to state the precise version they use explicitly in the proof rather than citing a chain. The other limitations are explicitly acknowledged in Section 1.6: the exponent Q/2(1+ε) is not optimal, and intermediate homological degrees are out of reach of this method. That is honest, not a hidden flaw.

Citation pattern: heavy self-citation is expected here; [14] is the Riemannian precursor and this paper is a direct extension. I don't see post-hoc fitting. The spectral step is the standard estimate ||f||_{W^{k,2}_X} ≤ C||(1+P)^{k/2}f||_2.

Who it's for: people working on nodal geometry, sub-Riemannian spectral theory, and topological persistence in analysis. It deserves a serious referee; the proofs are long but the structure is clear. My recommendation: engage with it, send it to a specialist on subelliptic PDE and a topologist who knows persistence. Minor revision, not a desk reject.

Referee Report

0 major / 4 minor

Summary. This paper proves coarse Courant- and B\'ezout-type bounds for finite linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds associated with stratified groups. The main analytic result is Theorem 1.5, a persistence-barcode estimate for $\pm|F|$ in homological degrees $0$ and $n-1$ in terms of the anisotropic Sobolev norm $\|F\|_{W^{k,p}_X}$, with exponent $Qp/(kp-Q)$. Through subelliptic coercivity (Theorem 4.17) and the spectral bound $\|(1+P)^{k/2}f\|_{L^2}\le(1+\lambda)^{k/2}\|f\|_{L^2}$, Theorems 1.1 and 1.3 follow, giving $(\lambda+1)^{Q/2(1+\epsilon_k)}\delta^{-2\epsilon_k}$ for $d\in\{0,n-1\}$. The proof of Theorem 1.5 combines ball-box geometry, a Vitali-Wiener covering, persistence subadditivity/stability, and the Lu-Wheeden Morrey-Sobolev polynomial approximation on Carnot balls. The paper also proves two non-optimal bounds in the general equiregular setting and formulates a conjecture with an explicit programme.

Significance. These appear to be the first results establishing the coarse Courant--Herrmann-type bound in the sub-Riemannian nilmanifold case, with an exponent arbitrarily close to the conjectured $Q/2$. The method is structurally transparent: the spectral statement is reduced to a barcode estimate for anisotropic Sobolev functions, and the only place where the stratified-group structure is essential is the ball-local polynomial approximation. The paper is explicit about its limitations---homological degrees only $0$ and $n-1$, a slightly worse exponent, and the absence of an equiregular analogue of Theorem 5.3. If the cited Lu-Wheeden theorem indeed holds in the uniform, ball-local form stated here, the proofs are internally consistent and the paper is a solid contribution to the subject.

minor comments (4)

[§5.3, Theorem 5.3] Theorem 5.3 is the single most load-bearing external input: the covering argument in the proof of Theorem 1.5 requires a constant independent of the center $g$, the radius $r$, and the chosen ball in the cover. The paper cites [38,39] and the first author's thesis [52], but does not reproduce the statement. Since a center-dependent constant would invalidate estimate (33), please either quote the precise theorem from the cited sources or add a short appendix containing the statement and the uniformity claim.
[§1.5 and §7] The sentence after Corollary 1.10 says that applying the corollary to $D=P^\kappa$ yields Theorems 1.1 and 1.3 only for even $k>Q/2$. However, the actual proof in Section 7 applies Theorem 1.5 directly with Theorem 4.17 and proves the statement for every integer $k>Q/2$. Please adjust the introductory remark so that it does not understate the range of $k$ covered.
[§6, proof of Theorem 1.5] In the trivial case at the start of the proof, the condition $C_0\|F\|_{W^{k,p}_X}<\delta$ can be weakened to $\le\delta$, since any finite bar has length at most $\max|F|-\min|F|\le\delta$. Also, the notation in (32), $N_{0,\delta}(\pm|F|_{\pi(B_\infty(g_i,r))}|)$, is slightly garbled; the intended meaning is clear but should be written cleanly.
[§2.3 and §6] The passage from degree $0$ to degree $n-1$ uses Proposition 2.20, which assumes $X$ is a closed, smooth, orientable manifold. Nilmanifolds of stratified groups are indeed orientable, but this fact is not stated where the $d=n-1$ case is treated. Adding one sentence would make the argument fully self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the core estimates are derived from an independent external approximation theorem and standard spectral inequalities.

full rationale

The derivation chain is: Theorems 1.1 and 1.3 are deduced from the Sobolev barcode estimate Theorem 1.5 (Section 7, equations (34)-(38)) via Proposition 2.21 and Theorem 4.17. Theorem 1.5 is proven in Section 6 by covering a fundamental domain with homogeneous balls, applying the Morrey–Sobolev polynomial approximation Theorem 5.3 (via Corollary 5.5), and then using barcode stability (Prop. 2.19) and subadditivity (Thm 2.17). Theorem 5.3 is quoted as a result of Lu–Wheeden [38, Thm 5.11; 39, Thm 5.11], with [52] only as a 'see also' appendix reference; it is an external, previously published theorem on stratified groups, not a restatement of the present target. The subadditivity and stability results are cited from [14], but they are general persistence facts proved there independently of any sub-Riemannian nodal bound. The spectral step (37)-(38) is the standard hypoelliptic coercivity estimate ||f||_{W^{k,2}_X} ≤ C||(1+P)^{k/2}f||_{L^2} followed by the eigenvalue bound (1+λ_j)^k ≤ (1+λ)^k; no fitted parameter is renamed as a prediction. The only acknowledged load-bearing external input is the Lu–Wheeden approximation theorem, which the paper explicitly identifies as the obstacle to the general equiregular conjecture (Section 1.6). No equation in the paper defines m_{d,δ} or z_{d,δ} in terms of the final bound, and no parameter is fitted to the data being 'predicted.' Thus there is no circular step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data, and no new entities are postulated. The central claim rests on standard persistence theorems, anisotropic Sobolev embedding, local polynomial approximation on Carnot groups, and subelliptic coercivity estimates, all cited to prior literature. The most fragile input is Theorem 5.3, whose equiregular analogue does not exist and is explicitly identified as the main obstacle to the broader conjecture.

assumptions (7)

standard math Barcode decomposition theorem for q-tame upper semi-continuous persistence modules (Theorem 2.7).
Needed to define N_{d,δ} and to connect coarse counts to bars; quoted from [50] and [22].
standard math Stability and subadditivity of barcode bar counts (Proposition 2.19, Theorem 2.17).
Core technical tool in the proof of Theorem 1.5; imported from persistence theory and [14].
standard math Poincare-duality-style barcode identity for finite bars on orientable closed manifolds (Proposition 2.20).
Used to pass from degree 0 to degree n−1 in Theorem 1.5; imported from [14].
domain assumption Lu-Wheeden local polynomial approximation on Carnot balls (Theorem 5.3).
Main analytic engine; gives C0 approximation by a polynomial with explicit scale-dependent error. The paper itself identifies the lack of an equiregular analogue as the main obstacle to the general conjecture.
domain assumption Anisotropic Sobolev embedding on nilmanifolds for kp>Q (Theorem 4.15).
Needed for sup-norm control and to ensure the oscillation of |F| is bounded by a multiple of the anisotropic Sobolev norm; from [48,56].
domain assumption Subelliptic coercivity for Hormander sub-Laplacians: ||f||_{W^{k,2}_X} ≤ C||(1+P)^{k/2}f||_{L^2} (Theorem 4.17).
Converts the Sobolev barcode estimate into eigenvalue-dependent bounds for eigenfunction combinations; from [55] and Hormander theory.
domain assumption Spectral theory of sub-Laplacians on compact nilmanifolds: discrete spectrum, smooth eigenfunctions, orthonormal basis, and Weyl-type law with homogeneous dimension Q.
Sets up F_λ and is used to bound ||(1+P)^{k/2}f||_{L^2}; standard from [24,40,49].

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Cite this review

Pith. "Pith review of Coarse nodal counts on sub-Riemannian manifolds." pith.science (2026). https://pith.science/paper/ENEOXBHB

@misc{pith2026260801446,
  author       = {Pith},
  title        = {Pith review of: Coarse nodal counts on sub-Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENEOXBHB}},
  note         = {Machine review of arXiv:2608.01446}
}

read the original abstract

We study coarse topology of nodal sets of linear combinations of eigenfunctions of sub-Laplacians. More precisely, we prove coarse versions of Courant's and B\'{e}zout's theorems for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds obtained as quotients of stratified groups. We conjecture the extensions of these results to general closed equiregular sub-Riemannian manifolds and outline a programme for proving them. The method we use combines topological persistence and anisotropic Sobolev theory of H\"{o}rmander vector fields, generalizing the ideas which have recently been implemented in the Riemannian case.

Figures

Figures reproduced from arXiv: 2608.01446 by the authors.

**Figure 1.** Coarse count of zeros Remark 1.4. Analogously to Theorem 1.1, Theorem 1.3 extends to sub-Laplacians associated to arbitrary measures with smooth positive densities and to general maximally hypoelliptic operators, see Remark 1.8, Corollary 1.10 and the discussion around it. 1.5. Anisotropic Sobolev estimates on barcodes. Let G = (G, ·) be a stratified group, g its stratified Lie algebra and let X1, . . . , Xm be a b… view at source ↗

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