pith. sign in

arxiv: 2412.20663 · v5 · submitted 2024-12-30 · 🧮 math.SP · math.DG

The hot spots conjecture on Gaussian spaces

Bobo Hua , Jin Sun This is my paper

Pith reviewed 2026-05-23 06:53 UTC · model grok-4.3

classification 🧮 math.SP math.DG
keywords hot spots conjectureGaussian spaceOrnstein-Uhlenbeck operatorLipschitz domainsmixed boundary conditionssymmetric domainseigenfunctionsHodge decomposition
0
0 comments X

The pith

The hot spots conjecture holds for Lipschitz domains with mixed boundary conditions and for n-symmetric domains in Gaussian spaces.

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

The paper proves that the first nontrivial eigenfunction of the Ornstein-Uhlenbeck operator on bounded domains in Gaussian space attains its extrema only on the boundary. This is shown for all Lipschitz domains under mixed Dirichlet-Neumann conditions and for n-symmetric domains where the intersection with one orthant is Lipschitz. A reader cares because the result identifies concrete families where the conjecture survives, even though it fails for some convex domains with other log-concave measures. The argument adapts the variational principle for the Hodge Laplacian and the Hodge decomposition of 1-forms from the Euclidean case to the Gaussian weighted setting.

Core claim

We establish the conjecture for lip domains in Gaussian spaces with mixed boundary conditions, and for n-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, any first nontrivial Neumann eigenfunction of a 2-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple.

What carries the argument

The variational principle for the Hodge Laplacian on weighted manifolds together with the Hodge decomposition of differential 1-forms on Lipschitz domains, extended to the Gaussian setting.

If this is right

  • The hot spots conjecture holds for all Lipschitz domains equipped with mixed boundary conditions.
  • The hot spots conjecture holds for all n-symmetric domains whose intersection with an orthant is Lipschitz.
  • In two-dimensional Gaussian space, every first nontrivial Neumann eigenfunction on a 2-symmetric domain has no interior extrema whenever the second Neumann eigenvalue is simple.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational technique may apply to other weighted manifolds or operators that admit a Hodge decomposition.
  • Symmetry plus Lipschitz regularity appears sufficient to force boundary extrema in this Gaussian setting.
  • Direct numerical computation of eigenfunctions on concrete 2-symmetric domains could test whether interior extrema are absent when the eigenvalue is simple.

Load-bearing premise

The variational principle and Hodge decomposition for the Ornstein-Uhlenbeck operator extend from the Euclidean setting to Gaussian space without additional restrictions on the weight or domain regularity beyond the stated Lipschitz and symmetry conditions.

What would settle it

An explicit first nontrivial eigenfunction on a Lipschitz domain with mixed boundary conditions or on an n-symmetric domain in Gaussian space that attains an extremum at an interior point would falsify the claim.

Figures

Figures reproduced from arXiv: 2412.20663 by Bobo Hua, Jin Sun.

Figure 1
Figure 1. Figure 1: Examples of lip domains Definition 1.3 (n-symmetric domain). A Lipschitz domain Ω ⊂ Mn is n-symmetric if Ω is invariant under reflection in each coordinate hyperplane {xk = 0}, k = 1, . . . , n. Given an n-symmetric domain Ω, we say that a subset O is an orthant of Ω if O = {x ∈ Ω : (−1)i1 x1 > 0, . . . ,(−1)in xn > 0} for some integers i1, . . . , in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of n-symmetric domains Remark 1.4. Due to the drift term −x · ∇ in the Ornstein–Uhlenbeck operator, the symmetry axes are required to pass through the origin. This differs slightly from the Euclidean setting treated in [27]. Definition 1.5 (Antisymmetric eigenfunction). Let Ω ⊂ Mn be an n-symmetric Lipschitz domain. An eigenfunction w of problem (1.2) with Σ = ∅ (pure Neumann conditions) is called… view at source ↗
read the original abstract

We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2\pi)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $\Omega$ with a piecewise smooth boundary, we consider the first nontrivial eigenfunction of the Ornstein--Uhlenbeck operator $L_\gamma = \Delta - \langle x, \nabla \rangle$ subject to Neumann or mixed Dirichlet--Neumann boundary conditions, and prove that its extrema are attained only on the boundary $\partial\Omega$. More precisely, we establish the conjecture for two classes of domains: (i) lip domains in Gaussian spaces with mixed boundary conditions, and (ii) $n$-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, we show that any first nontrivial Neumann eigenfunction of a $2$-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple. Our approach is based on a variational principle for the Hodge Laplacian on weighted manifolds and the Hodge decomposition of differential $1$-forms on Lipschitz domains, extending the variational method of Kennedy--Rohleder from the Euclidean setting to Gaussian spaces. Although de Dios Pont has shown that the hot spots conjecture can fail for certain convex domains endowed with suitable log-concave measures, our results identify broad classes of domains for which the conjecture remains valid in Gaussian spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes the hot spots conjecture for the first nontrivial eigenfunction of the Ornstein-Uhlenbeck operator L_γ = Δ − ⟨x, ∇⟩ on Gaussian space for two classes of domains: (i) Lipschitz domains with mixed Dirichlet-Neumann boundary conditions, and (ii) n-symmetric domains whose intersection with an orthant is Lipschitz. The proof extends the Kennedy-Rohleder variational method by invoking a variational principle for the Hodge Laplacian on weighted manifolds together with Hodge decomposition of 1-forms on Lipschitz domains.

Significance. If the extension of the method holds, the result identifies explicit classes of domains in Gaussian space where the conjecture remains valid, complementing known counterexamples for other log-concave measures and providing a weighted-space analogue of Euclidean hot-spots results.

major comments (1)
  1. [Approach paragraph] Approach paragraph (and the subsequent derivation of the variational principle): the manuscript asserts that the Hodge decomposition and integration-by-parts identities for the weighted Hodge Laplacian extend to the Ornstein-Uhlenbeck operator on the stated Lipschitz domains without further restrictions; however, the drift term ⟨x, ∇⟩ produces position-dependent contributions whose uniformity with respect to the location of Ω and the Lipschitz constant must be verified explicitly, as any implicit constant that grows with |x| would undermine the maximum-principle step for the first eigenfunction.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'piecewise smooth boundary' should be aligned with the 'lip domains' terminology used for the main theorems to avoid apparent inconsistency in regularity assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the approach. We address the point below.

read point-by-point responses

  1. Referee: Approach paragraph (and the subsequent derivation of the variational principle): the manuscript asserts that the Hodge decomposition and integration-by-parts identities for the weighted Hodge Laplacian extend to the Ornstein-Uhlenbeck operator on the stated Lipschitz domains without further restrictions; however, the drift term ⟨x, ∇⟩ produces position-dependent contributions whose uniformity with respect to the location of Ω and the Lipschitz constant must be verified explicitly, as any implicit constant that grows with |x| would undermine the maximum-principle step for the first eigenfunction.

    Authors: We thank the referee for this observation. All domains Ω in the manuscript are bounded, so there exists R < ∞ with |x| ≤ R on the closure of Ω. The drift term ⟨x, ∇⟩ is therefore bounded by R|∇| on Ω. The constants in the Hodge decomposition and integration-by-parts identities for the weighted Hodge Laplacian (with the smooth positive Gaussian weight) depend only on n, the Lipschitz constant of ∂Ω, and this R; they remain finite and uniform for any fixed domain. Consequently they do not grow with |x| in a manner that affects the maximum-principle argument for the first eigenfunction. We will insert an explicit verification of this uniformity in the approach paragraph of the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external Euclidean method

full rationale

The paper's central argument extends the variational principle and Hodge decomposition from Kennedy-Rohleder (Euclidean setting) to the Ornstein-Uhlenbeck operator on weighted Gaussian spaces for Lipschitz and symmetric domains. No quoted step reduces the target hot-spots result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the extension is presented as carrying over the external method under the stated regularity hypotheses. The abstract and approach paragraph identify the source of the method as prior independent work, with no renaming of known results or ansatz smuggling. This satisfies the self-contained criterion against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of the Hodge Laplacian and the validity of the variational principle when the underlying measure is Gaussian; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)
  • standard math Hodge decomposition holds for differential 1-forms on Lipschitz domains in weighted manifolds
    Invoked to extend the Euclidean variational method to the Gaussian setting.

pith-pipeline@v0.9.0 · 5789 in / 1205 out tokens · 30293 ms · 2026-05-23T06:53:36.228682+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 1 internal anchor

  1. [1]

    Differential Equations427(2025), 689–718

    Nausica Aldeghi and Jonathan Rohleder,On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions, J. Differential Equations427(2025), 689–718. MR4860866

    work page 2025

  2. [2]

    Ashbaugh, Fritz Gesztesy, Marius Mitrea, and Gerald Teschl,Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv

    Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, and Gerald Teschl,Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math.223(2010), no. 4, 1372–

    work page 2010

  3. [3]

    hot spots

    Rami Atar,Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem, Electron. J. Probab.6(2001), no. 18, 19. MR1873295

    work page 2001

  4. [4]

    Rami Atar and Krzysztof Burdzy,On nodal lines of Neumann eigenfunctions, Electron. Comm. Probab.7(2002), 129–139. MR1937899

    work page 2002

  5. [5]

    ,On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc.17(2004), no. 2, 243–265. MR2051611

    work page 2004

  6. [6]

    hot spots

    Rodrigo Ba˜ nuelos and Krzysztof Burdzy,On the “hot spots” conjecture of J. Rauch, J. Funct. Anal.164(1999), no. 1, 1–33. MR1694534

    work page 1999

  7. [7]

    hot-spots

    Rodrigo Ba˜ nuelos and Michael Pang,An inequality for potentials and the “hot-spots” con- jecture, Indiana Univ. Math. J.53(2004), no. 1, 35–47. MR2048182

    work page 2004

  8. [8]

    hot-spots

    Rodrigo Ba˜ nuelos, Michael Pang, and Mihai Pascu,Brownian motion with killing and reflec- tion and the “hot-spots” problem, Probab. Theory Related Fields130(2004), no. 1, 56–68. MR2092873

    work page 2004

  9. [9]

    hot spots

    Richard F. Bass and Krzysztof Burdzy,Fiber Brownian motion and the “hot spots” problem, Duke Math. J.105(2000), no. 1, 25–58. MR1788041

    work page 2000

  10. [10]

    Krzysztof Burdzy,The hot spots problem in planar domains with one hole, Duke Math. J. 129(2005), no. 3, 481–502. MR2169871

    work page 2005

  11. [11]

    hot spots

    Krzysztof Burdzy and Wendelin Werner,A counterexample to the “hot spots” conjecture, Ann. of Math. (2)149(1999), no. 1, 309–317. MR1680567

    work page 1999

  12. [12]

    Math.244(2026), no

    Hongbin Chen, Changfeng Gui, and Ruofei Yao,Uniqueness of critical points of the second Neumann eigenfunctions on triangles, Invent. Math.244(2026), no. 1, 299–353. MR5015432

    work page 2026

  13. [13]

    Hongbin Chen, Yi Li, and Lihe Wang,Monotone properties of the eigenfunction of Neumann problems, J. Math. Pures Appl. (9)130(2019), 112–129. MR4001630

    work page 2019

  14. [14]

    Haiyun Deng, Changfeng Gui, Xuyong Jiang, Xiaoping Yang, Ruofei Yao, and Jun Zou, Critical points of the second Neumann eigenfunctions on the quadrangles with symmetry, arXiv preprint arXiv:2604.19003 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  15. [15]

    Pedro Freitas,Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces, Indiana Univ. Math. J.51(2002), no. 2, 305–316. MR1909291

    work page 2002

  16. [16]

    Gernandt and J

    H. Gernandt and J. P. Pade,Schur reduction of trees and extremal entries of the Fiedler vector, Linear Algebra Appl.570(2019), 93–122. MR3912970

    work page 2019

  17. [17]

    Gol’dshtein, I

    V. Gol’dshtein, I. Mitrea, and M. Mitrea,Hodge decompositions with mixed boundary condi- tions and applications to partial differential equations on Lipschitz manifolds, 2011, pp. 347–

    work page 2011

  18. [18]

    Problems in mathematical analysis. No. 52. MR2839867

    work page

  19. [19]

    Lawford Hatcher,First mixed Laplace eigenfunctions with no hot spots, Proc. Amer. Math. Soc.152(2024), no. 12, 5191–5205. MR4855877

    work page 2024

  20. [20]

    ,The hot spots conjecture for some non-convex polygons, SIAM J. Math. Anal.57 (2025), no. 4, 4588–4608. MR4944076

    work page 2025

  21. [21]

    ,Hot spots in domains of constant curvature, arXiv preprint arXiv:2508.13353 (2025)

    work page arXiv 2025

  22. [22]

    ,Hot spots on cones and warped product manifolds, arXiv preprint arXiv:2508.18054 (2025)

    work page arXiv 2025

  23. [23]

    ,A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region, J. Spectr. Theory15(2025), no. 3, 1367–1382. MR4949380

    work page 2025

  24. [24]

    hot spots

    David Jerison and Nikolai Nadirashvili,The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc.13(2000), no. 4, 741–772. MR1775736

    work page 2000

  25. [25]

    Chris Judge and Sugata Mondal,Euclidean triangles have no hot spots, Ann. of Math. (2) 191(2020), no. 1, 167–211. MR4045963

    work page 2020

  26. [26]

    Partial Differential Equations47(2022), no

    ,Critical points of Laplace eigenfunctions on polygons, Comm. Partial Differential Equations47(2022), no. 8, 1559–1590. MR4462187

    work page 2022

  27. [27]

    Kennedy and Jonathan Rohleder,On the hot spots of quantum graphs, Commun

    James B. Kennedy and Jonathan Rohleder,On the hot spots of quantum graphs, Commun. Pure Appl. Anal.20(2021), no. 9, 3029–3063. MR4315494 18 BOBO HUA AND JIN SUN

    work page 2021

  28. [28]

    James B Kennedy and Jonathan Rohleder,On the hot spots conjecture in higher dimensions, arXiv preprint arXiv:2410.00816 (2025)

    work page arXiv 2025

  29. [29]

    Andreas Kleefeld,The hot spots conjecture can be false: some numerical examples, Adv. Comput. Math.47(2021), no. 6, Paper No. 85, 31. MR4349243

    work page 2021

  30. [30]

    Differential Equations266(2019), no

    David Krejˇ ciˇ r´ ık and Matˇ ej Tuˇ sek,Location of hot spots in thin curved strips, J. Differential Equations266(2019), no. 6, 2953–2977. MR3912674

    work page 2019

  31. [31]

    Lederman and Stefan Steinerberger,Extreme values of the Fiedler vector on trees, Linear Algebra Appl.703(2024), 528–555

    Roy R. Lederman and Stefan Steinerberger,Extreme values of the Fiedler vector on trees, Linear Algebra Appl.703(2024), 528–555. MR4806996

    work page 2024

  32. [32]

    2, 771–787

    Phanuel Mariano, Hugo Panzo, and Jing Wang,Improved upper bounds for the hot spots constant of Lipschitz domains, Potential Anal.59(2023), no. 2, 771–787. MR4625002

    work page 2023

  33. [33]

    hot spots

    Yasuhito Miyamoto,The “hot spots” conjecture for a certain class of planar convex domains, J. Math. Phys.50(2009), no. 10, 103530, 7. MR2572703

    work page 2009

  34. [34]

    hot spots

    ,A planar convex domain with many isolated “hot spots” on the boundary, Jpn. J. Ind. Appl. Math.30(2013), no. 1, 145–164. MR3022811

    work page 2013

  35. [35]

    Pascu,Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans

    Mihai N. Pascu,Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc.354(2002), no. 11, 4681–4702. MR1926894

    work page 2002

  36. [36]

    Jaume de Dios Pont,Convex sets can have interior hot spots, arXiv preprint arXiv:2412.06344 (2024)

    work page arXiv 2024

  37. [37]

    Jaume de Dios Pont, Alexander W Hsu, and Mitchell A Taylor,Sharp bounds on the failure of the hot spots conjecture, arXiv preprint arXiv:2508.16321 (2025)

    work page arXiv 2025

  38. [38]

    Jeffrey Rauch,Lecture# 1. five problems: An introduction to the qualitative theory of par- tial differential equations, Partial differential equations and related topics: Ford foundation sponsored program at tulane university, january to may, 1974, 1975, pp. 355–369

    work page 1974

  39. [39]

    Jonathan Rohleder,A variational approach to the hot spots conjecture, Extended abstracts 2021/2022—Methusalem lectures, [2024]©2024, pp. 37–45. MR4738461

    work page 2021

  40. [40]

    ,A note on hot-spots free subregions of convex domains, Complex Anal. Oper. Theory 20(2026), no. 2, Paper No. 57, 7. MR5030105

    work page 2026

  41. [41]

    1607, Springer-Verlag, Berlin, 1995

    G¨ unter Schwarz,Hodge decomposition—a method for solving boundary value problems, Lec- ture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR1367287

    work page 1995

  42. [42]

    Z.280(2015), no

    Bart lomiej Siudeja,Hot spots conjecture for a class of acute triangles, Math. Z.280(2015), no. 3-4, 783–806. MR3369351

    work page 2015

  43. [43]

    Partial Differential Equations45(2020), no

    Stefan Steinerberger,Hot spots in convex domains are in the tips (up to an inradius), Comm. Partial Differential Equations45(2020), no. 6, 641–654. MR4107000

    work page 2020

  44. [44]

    ,An upper bound on the hot spots constant, Rev. Mat. Iberoam.39(2023), no. 4, 1373–1386. MR4615892 Bobo Hua, School of Mathematical Sciences, Fudan University, 200433, Shanghai, China. Email address:bobohua@fudan.edu.cn Jin Sun, School of Mathematical Sciences, Fudan University, 200433, Shanghai, China. Email address:jsun22@m.fudan.edu.cn

    work page 2023