Sharp bounds on the failure of the hot spots conjecture

Alexander W. Hsu; Jaume de Dios Pont; Mitchell A. Taylor

REVIEW 2 major objections 3 minor 28 references

Sharp bounds on the failure of the hot spots conjecture

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves the exact maximum of the hot spots ratio in every dimension

desk verdict Big result, clean upper bound, but the lower-bound sieve scaling has an ε²-vs-ε slip that currently breaks the construction. read the letter →

arxiv 2508.16321 v1 pith:L4SVSETJ submitted 2025-08-22 math.SP math-phmath.APmath.MP

Jaume de Dios Pont , Alexander W. Hsu , Mitchell A. Taylor This is my paper · ORCID

classification math.SPmath-phmath.APmath.MP MSC 35P1535J0535B4058J50

keywords hotspotsconjectureNeumanneigenfunctionspectralgeometrysieverearrangementinequalityhigh-dimensionalasymptoticsBesselfunctionsquantitativestability

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 measures how badly Rauch's hot spots conjecture can fail by maximizing, over all connected Lipschitz domains, the ratio between a domain's interior hot spot and its hottest boundary point. It proves that this maximum is achieved asymptotically by a specific explicit function built from Bessel functions, namely the center value of the function η_d that solves the Helmholtz equation on the unit ball with boundary value 1. No domain actually reaches the maximum; every near-extremizer must be close to a ball, with a quantitative bound on its Fraenkel asymmetry. In high dimensions, the maximal ratio tends to √e, and the region of the domain where the eigenfunction exceeds its boundary maximum becomes exponentially small. The result turns a qualitative failure into a sharp, dimension-dependent quantity.

What carries the argument

The analysis is carried by η_d, an explicit radial eigenfunction of the Helmholtz equation on the unit ball with boundary value 1, written in terms of a Bessel function. The upper bound follows from a chain of inequalities: compare the normalized Neumann eigenfunction to the Dirichlet-type function u_{μ,Ω}, apply Talenti's rearrangement inequality to pass to the ball, then use monotonicity in the eigenvalue. The lower bound is produced by Neumann sieve domains—balls with a thin, highly perforated shell—whose effective first eigenfunction is radial and converges to η_d as the shell thins and the connectivity parameter approaches the ball's first nonzero Neumann eigenvalue. A key intermediate

What would settle it

Verify Proposition 20 numerically for a fixed dimension d: for each 0 < β < μ_{B_1(R^d)}, check whether the first positive eigenvalue h^(1)_{0,β,δ} of the radial effective problem is strictly smaller than the first eigenvalue h^(0)_{1,β,δ} of the non-radial ℓ=1 problem for all sufficiently small δ; if any β in that range yields h^(1)_{0,β,δ} > h^(0)_{1,β,δ} for a sequence δ→0, the lower-bound construction does not reach S_d.

Watch

Extended reading notes

Core claim

For every d ≥ 2, the supremum S_d of the hot spots ratio over connected Lipschitz domains is exactly η_d(0), where η_d is the radial function solving −Δη_d = μ_{B_1(R^d)}η_d on the unit ball with η_d = 1 on the boundary. No Lipschitz domain attains this value: extremizing sequences exist but must converge to a ball, and if a domain has unit-ball volume and its hot spots ratio is within ε² of S_d, then its Fraenkel asymmetry is at most C_d ε. As d → ∞, S_d converges to √e, matching the previously best known upper bound. The paper also determines the sharp function V_d(α), the largest possible measure of the set where the first Neumann eigenfunction exceeds α times its maximal boundary value;

Load-bearing premise

The lower bound relies on the claim that for every β below the ball's first nonzero Neumann eigenvalue, the first nontrivial mode of the effective sieve is radial once the sieve is thin enough; if that radiality fails, the constructed domains would not approach η_d.

Editorial extensions

If this is right

The hot spots ratio of any connected Lipschitz domain in dimension d is bounded by η_d(0), so the worst possible failure of the hot spots conjecture is now known exactly in every dimension.
Since no extremizer exists, the supremum can only be approached; the quantitative stability statement says that any domain whose ratio is within ε² of the supremum must be within O(ε) of a ball in Fraenkel asymmetry.
As d → ∞, the maximal ratio tends to √e, confirming that the previous upper bound was asymptotically sharp.
For any fixed threshold α > 1, the set where the first Neumann eigenfunction exceeds α times its boundary maximum has measure tending to zero exponentially fast as d → ∞, so the hot spots conjecture becomes 'true in measure' in high dimensions.
The sharp formula for V_d(α) gives a complete description of the distribution of super-level sets of the first Neumann eigenfunction, not just its L∞ norm.

Reading between the lines

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

The underlying mechanism suggests that interior hot spots arise only when the eigenfunction is forced to be radial by a weak disconnection; domain classes that rule out such sieves—for instance simply connected planar domains—may continue to satisfy the original conjecture.
The asymptotic value √e is the same one that appears from one-dimensional Gaussian marginals, hinting that high-dimensional spectral shape optimization collapses onto a Gaussian profile; it would be worth testing whether other spectral problems show the same dimensional reduction.
The paper's remark that in d ≥ 3 the holes can be connected to the boundary by capacity arguments makes it plausible that convex high-dimensional domains attain the same supremum; this is a testable route toward the paper's Conjecture 10.
The measure-theoretic bound likely transfers to L^p norms, giving quantitative hot-spots control in every L^p with exponentially small constants as the dimension grows.

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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

Let me save you time: this is the paper that claims to settle S_d exactly. The upper-bound half is a real piece of work: comparison (2.A), Talenti rearrangement (2.B), and monotonicity in μ (2.C) give S_d ≤ η_d(0), and the stability statement via Szego-Weinberger is credible. If that were all, it would still be a substantial paper. The asymptotic lim S_d = √e matching MPW is nice, and Corollary 9 (superlevel sets decay in measure) is a genuinely new observation.

The problem is the lower bound. The construction in Section 5 is supposed to converge to the effective form D_{β,δ}, whose jump penalty is βδ ∫_{S^{d-1}} J². But Definition 22 builds a spherical sieve with |S_ε| ≈ α ε² |S^{d-1}|. The necks have length ε and cross-sectional area |S_ε|, so their total conductance is |S_ε|/ε ≈ α ε |S^{d-1}|. That goes to zero, not to α. Therefore Lemma 25's first energy identity cannot hold as written: the neck contribution would be O(ε), not O(1), and the limiting form would have no jump penalty. The lower-bound sequence would collapse to a disconnected limit and not produce the sharp constant. This is load-bearing, not a cosmetic gap.

It also looks fixable, maybe a typo: if (5) used ε^{-1} instead of ε^{-2}, then |S_ε| ≈ α ε and the conductance is O(1). The homogenization estimates in Section 5 seem to tolerate that change; the exponential distribution argument should survive. But as submitted, the lower-bound half is not proved.

Two smaller issues: Proposition 20's spectral-decoupling proof is sketched—subsequential convergence plus Courant-Fischer is plausible but needs more detail before it is a proof. And Lemma 29 has a Fourier-transform/marginal mix-up (η_d is the characteristic function, not the density of the projected sphere measure); the limit is still e^{-r²/2}, so that one is cosmetic.

Bottom line: the upper bound and the conceptual framework are worth serious referee time, and I'd bet the result is true after the sieve exponent is corrected. But I would not desk-reject and I would not accept as-is. Send it out; the referee should be asked to check the scaling in Definition 22 and the proof of Proposition 20 carefully.

Referee Report

2 major / 3 minor

Summary. The paper studies the hot spots ratio S_d, the supremum over connected Lipschitz domains in R^d of the ratio of the maximum of the first nontrivial Neumann eigenfunction in the interior to its maximum on the boundary. The authors claim that S_d equals an explicit Bessel-type quantity η_d(0), that no extremizer exists, that extremizing sequences must converge to a ball at a quantitative rate, and that S_d → √e as d → ∞. They also prove a measure version V_d(α) and infer that the hot spots conjecture is asymptotically true in measure in high dimensions. The upper bound is obtained by a comparison principle, Talenti rearrangement, and monotonicity of the radial Dirichlet problem; the lower bound is constructed via a thick Neumann-sieve approximation of an effective two-parameter problem D_{β,δ}.

Significance. If correct, this would be a complete quantitative resolution of a well-known open-ended question around Rauch's hot spots conjecture, with matching asymptotics to the best known upper bound of Mariano–Panzo–Wang. The upper-bound argument is elegant and appears sound, and the stability statement is strong and explicit. The lower-bound Neumann-sieve construction is original and, if made fully rigorous, would be a substantial technical contribution. The paper is also transparent about the provenance of its ideas. However, the lower-bound homogenization as written contains a scaling error that invalidates the main equality claim in its present form.

major comments (2)

[§5, Definitions 22–23, Lemma 25, Proposition 24] The spherical sieve condition (5) implies |S_ε| ≈ α ε² |S^{d-1}|. The necks N_ε are radial tubes of length ε over S_ε. For a trace jump J, the minimal Dirichlet energy in N_ε is ≈ (J²/ε)|S_ε| = α ε |S^{d-1}| J², which tends to 0. The target form (3) contains βδ∫_{S^{d-1}}J² = α∫_{S^{d-1}}J², independent of ε. Hence the first identity in Lemma 25, and with it Proposition 24 and Proposition 17, cannot hold as stated: the sieve domains have vanishing connectivity, their nontrivial Neumann eigenvalue tends to 0, and the lower bound S_d ≥ η_d(0) is not established. The likely repair is to require |S_ε|≈α ε |S^{d-1}|, i.e. replace ε^{-2} by ε^{-1} in (5) and rerun the estimates of §5.2; but as submitted Theorem 4 is only an upper bound.
[§4.2, proof of Proposition 20] The spectral-decoupling argument is only sketched. The statements "Applying Courant-Fischer... spectrum decouples" and "the only interaction ... goes to zero" are asserted without the eigenvalue lower bounds and compactness needed to justify convergence of h^{(1)}_{0,β,δ} to β and the one-dimensional character of the outer limit. The sign pattern for r>1 is obtained after passing to a subsequence, with no proof of uniqueness of the subsequential limit (K_β is assumed to be -1). Since Proposition 20 is what guarantees radiality of ψ_{β,δ} and hence the applicability of the sieve construction and Proposition 18, this gap needs to be filled before the lower-bound claim can be accepted.

minor comments (3)

[Lemma 29] The sentence "c σ_d is a radial eigenfunction of the Laplace operator" is inaccurate because σ_d is a probability measure on a sphere, not a function. The intended statement is that the density of the projection π_{1*}σ_d (equivalently, the Fourier transform of σ_d) is proportional to \tilde η_d. The subsequent Gaussian limit is standard.
[Definition 22] The existence argument says "splitting the ball into roughly exp(−2/ε) pieces"; for balls of radius e^{-1/ε} on S^{d-1}, the relevant number of pieces is dimension-dependent, roughly exp(−(d−1)/ε). This should be corrected.
[Proof of Lemma 26] The displayed estimates contain garbled notation "ϵ− 1 ϵ" and "ϵ−2"; please reformat and check the exponents. As written the proof is hard to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the upper and lower bounds are independent, and self-citations are peripheral.

full rationale

The paper's central claim S_d = ||η_d||_∞ = η_d(0) is not circular. The upper bound (Section 2) proceeds through the chain ψ_Ω/max_∂Ω ψ_Ω ≤ u_{μ_Ω,Ω} ≤^♯ u_{μ_Ω,B_1} ≤ u_{μ_B1,B_1} = η_d, using the comparison principle, Talenti rearrangement, and monotonicity in μ. These are external standard tools; no parameter is fitted to the target value. The function η_d is defined purely from the ball's Dirichlet and Neumann spectral data, and the theorem derives its maximal value, rather than assuming it. The lower bound (Sections 3–5) is a fresh construction: a family of Neumann-sieve domains is introduced and shown to converge to an effective bilinear form D_{β,δ}; Proposition 20 analyzes the effective eigenfunction, and the limit β→μ_B1, δ→0 recovers η_d. This is an independent construction, not a renaming of the answer. The paper cites the first author's prior work [Dio24] only for folklore lower bounds and for motivating Conjecture 10, neither of which is load-bearing for Theorems 4 or 8; Theorem 4's proof does not reduce to that citation. No uniqueness theorem from the authors' prior work is invoked. The only substantive concern visible in the text is a possible scaling error in Definition 22/23 (the ε^{-2} volume fraction would make neck conductance O(ε), potentially invalidating Lemma 25 and Proposition 24). That is a correctness/falsifiability problem, not circularity: it does not make the claimed result equivalent to its inputs by construction. Therefore the circularity score is 0.

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

The central claim is parameter-free; beta, delta, and epsilon are internal construction parameters optimized in limits, not fitted to data. The proof invokes standard rearrangement theory, spectral stability, and sieve homogenization as background. No new physical or geometric objects are postulated beyond the mathematical construction itself.

free parameters (3)

beta (effective connectivity parameter) = n/a; optimized in the limit beta -> mu_{B_1(R^d)}
Introduced in the Neumann-sieve effective problem in (3). It is an auxiliary optimization parameter, not fitted to data.
delta (outer annulus thickness) = n/a; sent to 0
Parameter in the effective problem and sieve construction; the sharp lower bound is approached by taking delta -> 0.
epsilon (sieve scale) = n/a; sent to 0
Scale of the thick sieve in Section 5; the approximation to the effective problem is in the limit epsilon -> 0.

assumptions (5)

standard math Talenti's rearrangement inequality and the maximum principle for Laplace supersolutions
Used in the upper bound, Proposition 15, to compare u_{mu,Omega} with u_{mu,B_1(R^d)}.
standard math Szego-Weinberger eigenvalue bound mu_Omega <= mu_{B_1(R^d)} and the Brasco-Pratelli sharp stability estimate
Used in Section 2 for the upper bound and for the quantitative Fraenkel asymmetry convergence in Theorem 4.
domain assumption Existence of epsilon-spherical sieves satisfying the uniform distribution condition (5) via a probabilistic construction
Definition 22 in Section 5. The existence is asserted by splitting the sphere into exponentially small caps; the lower-bound construction depends on it.
standard math Bessel integral representation and high-dimensional Gaussian asymptotics for sphere measures
Invoked in Lemma 29, albeit with a misstated use of projection instead of Fourier transform. The underlying standard fact is that the Fourier transform of the uniform sphere measure of radius roughly sqrt(d) converges pointwise to a Gaussian in the relevant scaling.
standard math Uniqueness of radial solutions to the Helmholtz equation and monotonicity of u_{mu,B_1(R^d)} in mu
Used in Proposition 16 and in Section 4.2 to identify the radial eigenfunction with a Bessel function of fixed profile.

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Cite this review

Pith. "Pith review of Sharp bounds on the failure of the hot spots conjecture." pith.science (2026). https://pith.science/paper/L4SVSETJ

@misc{pith2026250816321,
  author       = {Pith},
  title        = {Pith review of: Sharp bounds on the failure of the hot spots conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4SVSETJ}},
  note         = {Machine review of arXiv:2508.16321}
}

abstract

The hot spots ratio of a domain $\Omega\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $\Omega\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$.

Figures

Figures reproduced from arXiv: 2508.16321 by the authors.

**Figure 1.** (Left) Plot of Sd as a function of d with the the asymptotic value Sd → √ e marked by a red line. (Right) Approximate values of Sd for different dimensions d along with the previously best known bounds. The value > 1 means that no specific value had been computed (to the knowledge of the authors), but the value is known to be > 1. Results marked with ∗ were folklore results. The fact that lim infd→∞ Sd > 1, without … view at source ↗

**Figure 2.** (Left) Sketch of the Neumann sieve domains. (Right) First Neumann Laplace eigenfunction for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** (Left) Hot spots ratio in dimension 3 as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The domain Ωϵ consists of B1+δ := B1+δ(R d ) with the region {x ∈ B1+δ s.t. |x| ∈ (1−ϵ/2, 1+ϵ/2)} removed along with thin channels through that region. We refer to the bulk as Wϵ and the channels as Nϵ. The width of the channels must be much smaller than their length, …

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

On N eumann eigenfunctions in lip domains

Rami Atar and Krzysztof Burdzy. On N eumann eigenfunctions in lip domains. Journal of the American Mathematical Society , 17(2):243--265, 2004

work page 2004

[2] [2]

On the hot spots conjecture of J

Rodrigo Ba \ n uelos and Krzysztof Burdzy. On the hot spots conjecture of J . R auch. Journal of Functional Analysis , 164(1):1--33, 1999

work page 1999

[3] [3]

The principal eigenvalue and maximum principle for second-order elliptic operators in general domains

Henri Berestycki, Louis Nirenberg, and SR Srinivasa Varadhan. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Communications on Pure and Applied Mathematics , 47(1):47--92, 1994

work page 1994

[4] [4]

Sharp stability of some spectral inequalities

Lorenzo Brasco and Aldo Pratelli. Sharp stability of some spectral inequalities. Geom. Funct. Anal. , 22(1):107--135, 2012

work page 2012

[5] [5]

The hot spots problem in planar domains with one hole

Krzysztof Burdzy. The hot spots problem in planar domains with one hole. Duke Mathematical Journal , 129:481--502, 2005

work page 2005

[6] [6]

A counterexample to the `` H ot spots" conjecture

Krzysztof Burdzy and Wendelin Werner. A counterexample to the `` H ot spots" conjecture. Annals of Mathematics , pages 309--317, 1999

work page 1999

[7] [7]

Monotone properties of the eigenfunction of N eumann problems

Hongbin Chen, Yi Li, and Lihe Wang. Monotone properties of the eigenfunction of N eumann problems. Journal de Math \'e matiques Pures et Appliqu \'e es , 130:112--129, 2019

work page 2019

[8] [8]

Le probleme de la passoire de N eumann

A Damlamian. Le probleme de la passoire de N eumann. Rend. Sem. Mat. Univ. Politec. Torino , 43(3):427--450, 1985

work page 1985