Maximizing higher eigenvalues in dimensions three and above

Denis Vinokurov

REVIEW 3 major objections 4 minor 3 cited by

Maximizing higher eigenvalues in dimensions three and above

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that for any closed Riemannian manifold of dimension $m\ge3$ and any $k\ge1$, the supremum $V_k(g)$ of the $k$-th Laplace eigenvalue over measures is attained by the energy measure of a locally stable harmonic map into…

desk verdict Genuinely new existence result for all k in dims ≥3, with a plausible but under-written regularity section that referees should push on. read the letter →

arxiv 2506.09328 v3 pith:PO3AU7UE submitted 2025-06-11 math.SP math.APmath.DG

Denis Vinokurov This is my paper · ORCID

classification math.SPmath.APmath.DG MSC 58J5035P1558E20

keywords eigenvalueoptimizationharmonicmapsabsolutelycontinuousmeasuresLaplaceeigenvaluessingularsetℓ²sphereHausdorffdimensionStiefel–Whitneyclasses

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

The paper proves a higher-dimensional existence theorem for eigenvalue optimization: on any closed Riemannian manifold of dimension $m\ge3$, and for every $k\ge1$, the supremum $V_k(g)$ of the $k$-th Laplace eigenvalue over absolutely continuous measures is attained. The maximizing measure is always the energy density $|du|^2\,dv_g$ of a locally stable harmonic map into the unit sphere of $\ell^2$, smooth away from a closed singular set of Hausdorff dimension at most $m-7$. In dimensions $3\le m\le6$ the map is smooth and lands in a finite-dimensional sphere, so every such manifold carries infinitely many smooth harmonic maps that are eigenvalue maximizers. The paper also shows the $m-7$ bound is sharp by explicit generalized equator maps on round spheres whose singular sets have any prescribed dimension up to $m-7$. Why care: it settles existence and describes the regularity of extremal measures for all Laplace eigenvalues in all dimensions at once.

What carries the argument

Two mechanisms carry the argument. First, the paper works with nuclear (trace-class) tensors $t_\varepsilon=\sum_i \varphi_i^\varepsilon\otimes\varphi_i^\varepsilon\in\mathcal{N}[H^1(M)]$ instead of vector-valued maps; the weak-* topology on $\mathcal{N}[H^1]$ is strong enough that any limit $t=\sum_i\varphi_i\otimes\varphi_i$ satisfies $\sum_i|\varphi_i|^2=1$, exactly the normalization that fails for the non-compact embedding $H^1(M,\ell^2)\hookrightarrow L^2(M,\ell^2)$. Second, the nonsmooth calculus of eigenvalue functionals on compact self-adjoint operators, through Clarke and approximate subdifferentials, controls how many eigenfunctions can appear and yields the spherical eigenmap. Regularity is then governed by the stability inequality $\int \omega^2|du|^2\,dv_g\le\int|d\omega|^2\,dv_g$ for harmonic maps into $S^\infty$, which, together with small-energy regularity and dimension reduction, confines the singular set to Hausdorff dimension at most $m-7$.

What would settle it

Find a nonconstant stable harmonic map $\varphi(x/|x|):\mathbb{R}^{d+1}\to S^\infty$ for some $3\le d+1\le6$; Theorem 5.14's dimension-reduction argument asserts no such tangent map exists, so constructing one would contradict the claimed $m-7$ bound.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for every closed Riemannian manifold $(M,g)$ of dimension $m\ge3$ and every $k\ge1$, the supremum $V_k(g)=\sup_\mu \lambda_k(\mu)\mu(M)$ over absolutely continuous measures is attained by the energy measure $|du|^2\,dv_g$ of a locally stable harmonic map $u\in H^1(M,S^\infty)$ into the unit sphere of $\ell^2$; the map is smooth outside a closed set $\Sigma$ of Hausdorff dimension at most $m-7$, and in dimensions $3\le m\le6$ it is smooth and takes values in a finite-dimensional sphere. The paper further proves the bound is sharp: for every $m\ge7$ and every integer $0\le d\le m-7$ there is a maximizing harmonic map on the round sphere whose singular set has Hausdorff dimension exactly $d$.

Load-bearing premise

The proof asks the reader to grant that stable harmonic maps into the infinite-dimensional sphere obey the same small-energy regularity and tangent-map classification as maps into finite-dimensional spheres, and if that fails, smoothness in dimensions 3–6 and the m−7 bound collapse.

Editorial extensions

If this is right

For every closed $m$-manifold and every $k\ge1$, $V_k(g)$ is attained, and the extremal measure is the energy measure of a locally stable harmonic map; no atom can appear in a maximizing sequence when $m\ge3$.
In dimensions $3\le m\le6$, every closed manifold carries infinitely many smooth harmonic maps into finite-dimensional spheres, each realizing some $V_k(g)$.
The Hausdorff-dimension bound $m-7$ is optimal: on the round $m$-sphere, $m\ge7$, generalized equator maps give maximizers with singular sets of any dimension $0\le d\le m-7$, and for $k\le m-7$ the equator map is the unique maximizer of $V_{k+2}$.
An analogue holds for Steklov eigenvalues on manifolds with boundary: the supremum is attained by a locally stable free-boundary harmonic map into the unit ball of $\ell^2$.
The maximizing sequence exhibits no bubbling, so concentration of energy, the obstruction in surface eigenvalue optimization, disappears in dimensions $m\ge3$.

Reading between the lines

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

If the theorem is right, the extremal objects in spectral optimization are exactly the finite-index harmonic maps to spheres; this gives a two-way bridge between eigenvalue extremals and harmonic-map theory that could let spectral questions be attacked with geometric-analysis tools.
The paper leaves open whether $S^\infty$ is ever needed (Questions 1.10–1.11). A natural next step is to try to construct a finite-index harmonic map to $S^\infty$ with infinitely many nonzero coordinate functions; if none exists, all maximizers are finite-dimensional.
The same nuclear-operator scheme should apply to the $p$-harmonic family $V_{k,q}$; in the borderline case $q=m/(m-2)$, establishing existence of an $m$-harmonic map maximizer would imply the equality $V_{k,m/(m-2)}(g)=\Lambda_k([g])$, connecting to conformal eigenvalue optimization.
Because atoms are ruled out in $m\ge3$, the theorem predicts a concrete numerical signature: fine-grid computations of maximizing measures should show no concentration, unlike the two-dimensional case.

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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

The headline: this paper genuinely breaks new ground. Vinokurov proves existence of maximizers for every k-th eigenvalue functional V_k on closed manifolds of dimension m≥3, realizing the supremum via harmonic maps into S^∞ with a sharp m-7 singular-set bound, and shows the bound is attained on round spheres by generalized equator maps. That is a real advance beyond Karpukhin–Stern (first eigenvalue, dims 3–5) and the 2D case. The nuclear-operator/tensor framework is a clever and apparently workable way around the non-compactness of H^1(M,ℓ²). I read Sections 1–4 carefully; the main line is coherent. The maximizing-sequence construction, the no-bubbling result in m≥3, the convergence to a stable harmonic map, and the index computation for equator maps are all substantial and mostly well-written. Theorem 1.8's index count is explicit, and I did not find a gap.

The soft spots are concentrated in Section 5. Theorem 5.14 is load-bearing: it supplies the m-7 bound and smoothness in dims 3–6. Its proof is a concise adaptation of Hong–Wang and Schoen–Uhlenbeck. Lemma 5.13, which gives strong N[H^1] convergence for bounded sequences of stable S^∞-valued harmonic maps, is dispatched in a paragraph that essentially says “apply Lemmas 4.2 and 4.3 with D=∅.” As far as I can tell the application is legitimate: stability exactly gives ind=0, [t_ε]=[t]=1, and the equations match. But the argument is compressed, and the dimension-reduction step that leads to m-7 is one sentence citing [SU1, Section 5] and [HW, Lemma 2.3]. For an ℓ²-valued target one needs the Federer slicing argument to go through; the paper gestures at this rather than proving it. I would not call it fatal—the adaptation is plausible and the missing details look standard—but a referee should demand a written-out proof of Lemma 5.13 and the dimension reduction. Theorem 4.7 (Steklov) is also only sketched; it is secondary, but if claimed, it should be proved or clearly marked as a follow-up.

One more thing: the arXiv metadata abstract says the maximizer is a harmonic map into a finite-dimensional sphere for every m, which contradicts Theorem 1.1 (S^∞ for m≥7; finite-dimensional only for m≤6). The paper's own abstract is consistent; the metadata is wrong. The authors should fix that before it causes confusion.

Who this is for: spectral geometers and anyone working on extremal eigenvalue problems or harmonic maps into infinite-dimensional targets. It deserves a serious referee and, after the Section 5 details are tightened, publication in a top journal.

Referee Report

3 major / 4 minor

Summary. The paper studies the optimization problem V_k(g)=sup_{µ} λ_k(µ) µ(M), where the supremum is taken over absolutely continuous measures on a closed Riemannian manifold of dimension m≥3. The main theorem (Theorem 1.1) asserts that for every k≥1 the supremum is attained by the energy measure of a locally stable harmonic map u∈H^1(M,S^∞), that this map satisfies λ_k(|du|²dv_g)=1, and that it is smooth outside a closed singular set of Hausdorff dimension at most m−7; in particular it is smooth for 3≤m≤6 and then takes values in a finite-dimensional sphere (Corollary 1.3). The paper also analyzes the round sphere: Theorem 1.6 gives an upper bound for λ_{k+2} by the energy of generalized equator maps, and Theorem 1.8 computes their spectral index and shows that for 0≤k≤m−7 they realize V_{k+2}(g_{S^m}), yielding singular maximizers with singular sets of any dimension up to m−7. The proofs combine a variational construction using nuclear operators and Clarke/approximate subdifferentials, a topological argument on Grassmannians, a no-bubbling result, and a partial-regularity theory for harmonic maps into the infinite-dimensional sphere S^∞.

Significance. If the main results are correct, the paper solves the existence problem for maximizers of higher Laplace eigenvalues over measures in all dimensions m≥3 and establishes an optimal dimension bound m−7 for their singular set. This is a substantial extension of the earlier work of Karpukhin and Stern, and the explicit construction of sphere maximizers with prescribed singular-set dimensions is a valuable contribution. The variational machinery involving nuclear operators and approximate subdifferentials is innovative and is developed in a largely self-contained way. The paper also gives concrete predictions, such as the uniqueness of the generalized equator maps as maximizers of V_{k+2} on the round sphere, and it poses clear open questions. The main weakness is that the infinite-dimensional partial-regularity theorem (Theorem 5.14), on which the smoothness and the dimension bound rest, is proved in a highly compressed style: its key compactness lemma and the dimension-reduction step are reduced to earlier lemmas and external citations without a fully detailed verification.

major comments (3)

[§5, Lemma 5.13 and Theorem 5.14]
[§5, Theorem 5.14]
[Corollary 1.3, proof]

minor comments (4)

[Abstract]
[§5, Theorem 5.14]
[Throughout]
[§5, Lemma 5.12]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proof is self-contained; the only self-citation [Vin] supports auxiliary, parameter-free lemmas and does not force the main result.

full rationale

The paper's central claim, Theorem 1.1, is derived from an independently defined quantity V_k(g) in (1.4). The proof constructs a maximizing sequence of measures and eigenfunction tensors t_ε in N[H^1], passes to a weak* limit, and uses the identity [t]=1 (equation (4.8)) to extract a sphere-valued map u. The harmonic map equation and the measure identity μ=|du|^2 dv_g follow from the eigenvalue equation, stability inequality (2.1), and standard capacity arguments, none of which assume the conclusion. The no-atoms step uses the bubbling Proposition 4.6, whose proof rescales around potential atoms and is independent of the target theorem. The regularity input Theorem 5.14 adapts external results [HW], [SU1], [SU2], [LW], [Pri], with the infinite-dimensional compactness supplied by the paper's own nuclear-operator lemmas (Lemmas 2.12, 4.2, 4.3), which are proven from general operator theory. The only self-citation, [Vin], appears for the bubbling lemma (Prop 4.5), an alternative proof of the Clarke subdifferential characterization (Prop 2.18), and the surface-case adaptation; each is a parameter-free auxiliary statement whose assumptions do not include Theorem 1.1. No fitted parameter is introduced, no eigenvalue is predicted from a fit, and no step reduces by construction to its own input. The manuscript itself flags the compressed nature of parts of Section 5 (e.g., the dimension-reduction argument), but that is a proof-detail gap, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure mathematical proof. It introduces no new physical entities and no numbers fitted to data. All substantive inputs are standard results in spectral geometry, harmonic map regularity, nonsmooth analysis, and algebraic topology, which are cited and applied. The theoretical framework with nuclear operators is a new method but not an ad hoc entity; it is developed rigorously in Sections 2 and 4.

assumptions (4)

standard math Background partial regularity for harmonic maps into spheres: monotonicity formula (Price), small-energy regularity (Chang-Wang-Yang), dimension reduction (Schoen-Uhlenbeck) and stable stationary regularity (Hong-Wang) apply to S^∞-valued maps via Banach-valued Sobolev theory.
Invoked in Section 5 to prove Theorem 5.14 (partial regularity for stable S^∞-valued harmonic maps), which is the basis for the m-7 singular set bound and the smoothness in dimensions 3-6.
standard math Compactness of the embedding H^1(M) ↪→ L^2(M) and spectral theory of the Laplace-Beltrami operator on a closed manifold, including the variational characterization of eigenvalues and the compactness of the associated resolvent.
Used in Section 2.1 (Prop. 2.6, Prop. 2.7) and Corollary 2.15 to justify the nuclear-operator framework and the weak-to-strong convergence of tensors.
standard math Nonsmooth analysis on Banach spaces: Clarke subdifferential, approximate subdifferential, chain rules and the Lagrange multiplier rule for locally Lipschitz functionals.
Used in Sections 2.3-2.4 (Prop. 2.17, 2.18, 2.21, 2.22) to obtain the first variation of λ_k and the existence of the eigenmap in Corollary 2.23.
standard math Stiefel-Whitney class computations for the tautological bundle over the Grassmannian: the top Stiefel-Whitney class of the (k+1)-fold Whitney sum is nonzero (Lemma 3.1, from Stong and Korbaš-Novotný).
Used in the proof of Theorem 1.6 to guarantee a vanishing continuous section, giving the orthogonality conditions for the test maps; this is a standard topological input.

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Pith. "Pith review of Maximizing higher eigenvalues in dimensions three and above." pith.science (2026). https://pith.science/paper/PO3AU7UE

@misc{pith2026250609328,
  author       = {Pith},
  title        = {Pith review of: Maximizing higher eigenvalues in dimensions three and above},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO3AU7UE}},
  note         = {Machine review of arXiv:2506.09328}
}

abstract

We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. Extending the work of Karpukhin and Stern on the first eigenvalue, we prove that, for every $k\geq 1$, the supremum is attained by a measure induced by a harmonic map into a finite-dimensional sphere. The map is smooth outside a closed singular set of Hausdorff dimension at most $m-7$, and is therefore smooth when $3 \leq m \leq 6$. We further prove that this dimension bound is optimal: for every $m \geq 7$ and every integer $0\leq d \leq m-7$, there exists a maximizing harmonic map on the $m$-dimensional round sphere whose singular set has Hausdorff dimension $d$.

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Reviewed August 7, 2026 · model on record in the stance chip above.

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[P´ er2] S. P´ erez-Ayala. “Extremal metrics for the Paneitz operator on closed four-manifolds”. In: J. Geom. Phys. 182 (2022), Paper No. 104666,

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A monotonicity formula for Yang-Mills fields

arXiv: 2410.13347v1 [math.DG]. [Pri] P. Price. “A monotonicity formula for Yang-Mills fields”. In: manuscripta mathematica 43.2 (June 1, 1983), pp. 131–166. [PT] R. Petrides and D. Tewodrose. Critical metrics of eigenvalue functionals via Clarke subdifferential

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arXiv: 2403.07841v1 [math.DG]. [Sch] W. Schirotzek. Nonsmooth analysis. Universitext. Springer, Berlin, 2007, pp. xii+373. [Sto] R. E. Stong. “Cup products in Grassmannians”. In: Topology and its Applications 13.1 (Jan. 1, 1982), pp. 103–113. [SU1] R. Schoen and K. Uhlenbeck. “A regularity theory for harmonic maps”. In: Journal of Differential Geometry 17...

arXiv 2007

[8] [24]

Surv. Differ. Geom. Int. Press, Boston, MA, 2022, pp. 205–256. [KNPS] M. Karpukhin, M. Nahon, I. Polterovich, and D. Stern. Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces

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