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arxiv: 2604.19003 · v1 · submitted 2026-04-21 · 🧮 math.AP

Critical points of the second Neumann eigenfunctions on the quadrangles with symmetry

Haiyun Deng , Changfeng Gui , Xuyong Jiang , Xiaoping Yang , Ruofei Yao , Jun Zou This is my paper

Pith reviewed 2026-05-10 02:42 UTC · model grok-4.3

classification 🧮 math.AP
keywords Neumann Laplacianeigenfunctionssymmetrycritical pointsHot Spots Conjecturetrapezoidsparallelogramskites
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The pith

The second Neumann eigenfunction on isosceles trapezoids, parallelograms and kites has symmetry properties that depend on angles and heights and fully determine its critical points.

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 second Neumann eigenfunction u on an isosceles trapezoid is antisymmetric about the symmetry axis when the base angle α is at most π/3. When α exceeds π/3, a critical height ĥ exists such that u is antisymmetric for heights below ĥ and symmetric above it, with the eigenvalue having multiplicity two exactly at ĥ. For parallelograms, u is always centrally antisymmetric about the center and has no critical points away from vertices. For kite domains, the symmetry about the x-axis switches from symmetric to antisymmetric as height increases past two critical values h0 and h1 when the parameter a is less than 2. These results include a complete description of where any non-vertex critical points of u lie on the closed domain and give affirmative answers to the Hot Spots Conjecture for all three families of quadrilaterals.

Core claim

When Q is an isosceles trapezoid with base angle α, if α ≤ π/3 then the second Neumann eigenfunction u is antisymmetric about the symmetric axis. If α > π/3 there is a critical height ĥ so that u is antisymmetric for h < ĥ, symmetric for h > ĥ, and the multiplicity is 2 at h = ĥ. When Q is a parallelogram u is centrally antisymmetric about the center of Q and has no non-vertex critical points; in the rhombus case it is symmetric with respect to the longer diagonal and antisymmetric with respect to the shorter diagonal. When Q is a kite with parameters a and h, if a ≥ 2 then u is antisymmetric about the x-axis, while if 0 < a < 2 there are h0 ≤ h1 such that u is symmetric for h < h0 and antis

What carries the argument

Symmetry decomposition of the eigenfunction together with eigenvalue comparisons and the continuity method applied to varying heights and angles.

If this is right

  • The non-vertex critical points of u lie only on the boundaries or symmetry axes consistent with the established symmetry.
  • The Hot Spots Conjecture holds for isosceles trapezoids, parallelograms and kites.
  • When the domain is a rhombus, u is symmetric with respect to the longer diagonal and antisymmetric with respect to the shorter diagonal.
  • At the identified critical heights the second eigenvalue has multiplicity exactly two.

Where Pith is reading between the lines

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

  • These symmetry transitions may indicate a change in the nodal set structure of the eigenfunction at the critical parameters.
  • The method could be extended to other quadrilaterals or polygons with one or two varying parameters to find similar critical values.
  • The absence of interior critical points in parallelograms suggests that for convex domains with central symmetry the second eigenfunction has no interior maxima or minima.

Load-bearing premise

The continuity method tracks the second eigenfunction without it losing simplicity or crossing with other eigenvalues in a way that would prevent the symmetry classification throughout the parameter ranges.

What would settle it

A numerical or analytical calculation of the second eigenfunction on a specific isosceles trapezoid with base angle greater than π/3 and height larger than the critical value, verifying whether it is symmetric or antisymmetric about the axis.

Figures

Figures reproduced from arXiv: 2604.19003 by Changfeng Gui, Haiyun Deng, Jun Zou, Ruofei Yao, Xiaoping Yang, Xuyong Jiang.

Figure 1
Figure 1. Figure 1: Geometric distribution of level sets of u in isosceles trapezoidal domains (α > π 3 ). (−1, 0), Q2 = (1, 0), Q3 = [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometric distribution of level sets of u in isosceles trapezoidal domains (α < π 3 ). The following results concern the hot spots conjecture on parallelograms and kite domains. Theorem 1.2. (1) If Q is the parallelogram P1P2P3P4 with |P1P2| ≥ |P1P4|, where P1 is the origin, P2 is on the positive x-axis, P3 and P4 lie in the first quadrant. Let C be the center of Q. Then the second Neumann eigenfunction u … view at source ↗
Figure 3
Figure 3. Figure 3: Geometric distribution of level sets of u in the parallelogram domains. Theorem 1.3. Let K be the kite P1P2P3P4, where P1 is the origin, P2 = (a, −h) lies in the four quadrant, P3 = (1, 0) lies on the positive x-axis, and P4 = (a, h) is symmetric with P2 about x-axis which lies in the first quadrant. Then (1) If a ≥ 2, then the second Neumann eigenfunctions are antisymmetric about x-axis. (2) If 0 < a < 2,… view at source ↗
Figure 4
Figure 4. Figure 4: Geometric distribution of level sets of u in the Kite domains. The geometric distribution of level sets of the second Neumann eigenfunctions u is shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Geometric distribution of level sets of u in the Kite domains. Case 3 (a > 2). Let a = 2.10. In this case, we consider two kites of heights 0.65 and 0.5, respectively. The geometric distribution of level sets of the second Neumann eigenfunctions u is shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Geometric distribution of level sets of u in the Kite domains. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: ∠P2P1P4 ≥ π 2 , ∠P2P3P4 ≥ π 2 , ∠P1P4P3 ≤ π 2 and ∠P1P4P3 + ∠P2P3P4 ≥ π. (1) The second Neumann eigenvalue is simple. (2) u does not have any non-vertex critical points, and u is monotonic along the four edge directions. (3) u attains its global extrema at P2 and P4, P1 and P3 are saddle points. Remark 3.4. Since the parallelogram satisfies the above requirements, then the second Neumann eigenvalue is simp… view at source ↗
Figure 9
Figure 9. Figure 9: Geometric distribution of level sets of u in the right trapezoidal domain. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Geometric distribution of level sets of u in the right trapezoidal domain with mixed boundary conditions. Lemma 3.10. Let Qh be the right trapezoid P1P2P3P4 with height h, where P1 is the origin, P2 is a fixed point on the x-axis, P3 lies in the first quadrant, P4 lies on the y-axis and the base acute angle ∠P1P2P3 is fixed. Let λh be the first eigenvalue of the mixed boundary problem    ∆u + λu … view at source ↗
Figure 11
Figure 11. Figure 11: Limit of the second Neumann eigenvalues and limit of the first mixed [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗
read the original abstract

In this paper, we focus primarily on the symmetry properties of the second Neumann eigenfunction $u$ with respect to the symmetry axis or symmetry center of the relevant domain $Q$, such as isosceles trapezoids, parallelograms, kite domains, and we provide some affirmative answers to the Hot Spots Conjecture for these domains. Our proofs combine symmetry decomposition, comparison of eigenvalues, and the continuity method. Precisely, we have the following three aspects of results. (1) when $Q$ is an isosceles trapezoid, if the base angle $\alpha\le \frac{\pi}{3}$, $u$ is antisymmetric about the symmetric axis; if the base angle $\alpha> \frac{\pi}{3}$, there exists a critical height $\hat{h}$, when height $h<\hat{h}$, $u$ is antisymmetric about the symmetric axis; when height $h>\hat{h}$, $u$ is symmetric about the symmetric axis; when height $h=\hat{h}$, the multiplicity of second Neumann eigenvalue is 2. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$. (2) When $Q$ is a parallelogram, $u$ is centrally antisymmetric about the center of $Q$ and does not have any non-vertex critical points. In particular, when $Q$ is a rhombus, $u$ is symmetric with respect to the longer diagonal and is antisymmetric with respect to the short diagonal. (3) When $Q$ is a kite $P_1P_2P_3P_4$, where $P_1$ is the origin, $P_2=(a,-h)$ lies in the four quadrant, $P_3=(1,0)$ lies on the positive $x$-axis, and $P_4=(a,h)$ is symmetric with $P_2$ about $x$-axis which lies in the first quadrant. If $a\ge 2$, $u$ is antisymmetric about $x$-axis; if $0<a<2$, there exist two constants $h_0$ and $h_1$ ($h_0\le h_1$), when $h<h_0$, $u$ is symmetric about $x$-axis; when $h>h_1$, $u$ is antisymmetric about $x$-axis. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$.

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

3 major / 2 minor

Summary. The manuscript examines symmetry properties of the second Neumann eigenfunction u on symmetric quadrilaterals (isosceles trapezoids, parallelograms, kites) and locates non-vertex critical points, providing affirmative cases for the Hot Spots Conjecture. For isosceles trapezoids with base angle α > π/3, it identifies a critical height ĥ at which symmetry of u switches from antisymmetric to symmetric about the axis (with multiplicity 2 at ĥ); for α ≤ π/3, u is always antisymmetric. For parallelograms, u is centrally antisymmetric with no non-vertex critical points (and additional diagonal symmetries for rhombi). For kites with parameter a, symmetry about the x-axis switches between symmetric and antisymmetric regimes separated by h0 ≤ h1. Proofs combine symmetry decomposition into even/odd modes, eigenvalue comparisons, and the continuity method in the height parameter.

Significance. If the claims hold, the results give explicit parameter-dependent symmetry classifications and critical-point locations for the second Neumann eigenfunction on these families of quadrilaterals, extending known affirmative cases of the Hot Spots Conjecture. The combination of symmetry reduction with a continuity argument that detects a symmetry switch at an isolated multiplicity-2 point is a technically interesting approach; the complete characterization of non-vertex critical points on the closure is a concrete contribution.

major comments (3)
  1. [continuity method for isosceles trapezoids] In the continuity argument for isosceles trapezoids (aspect (1) of the abstract), the manuscript asserts that the second eigenvalue is simple except at the isolated critical height ĥ where multiplicity equals 2, and that the symmetry type of a continuous branch of u switches exactly from antisymmetric (h < ĥ) to symmetric (h > ĥ). Because a two-dimensional eigenspace at ĥ admits linear combinations with mixed symmetry, the argument must explicitly rule out additional crossings of the symmetric and antisymmetric branches elsewhere in the interval and confirm that the ordering of the first two eigenvalues remains consistent with the claimed simplicity for h ≠ ĥ. This justification is load-bearing for the symmetry classification.
  2. [continuity method for kites] The same issue arises for kite domains (aspect (3)): the statements that u is symmetric for h < h0 and antisymmetric for h > h1, with possible multiplicity-2 points at h0 and h1, require a detailed verification that the second eigenvalue stays simple away from those isolated values and that no further intersections occur between the even and odd branches in the parameter ranges 0 < a < 2. Without this, the claimed switch in symmetry type cannot be guaranteed for the full interval.
  3. [critical-point location arguments] The location of non-vertex critical points on the closure (claimed for both trapezoids and kites) relies on the symmetry classification together with maximum-principle or nodal-domain arguments. These steps must be shown to remain valid at the multiplicity-2 points and for all parameter values in the stated ranges; any dependence on the eigenfunction being strictly symmetric or antisymmetric needs explicit justification when multiplicity occurs.
minor comments (2)
  1. [kite domain definition] Notation for the kite vertices (P1 at origin, P2=(a,-h), etc.) should be introduced with a figure or explicit coordinate diagram to avoid ambiguity when a and h vary.
  2. [method overview] The manuscript should clarify whether the continuity method is applied after fixing the first eigenvalue or after the symmetry decomposition; a brief remark on how the variational characterization interacts with the parameter deformation would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below and will incorporate the requested clarifications and additional justifications into the revised version.

read point-by-point responses

  1. Referee: In the continuity argument for isosceles trapezoids (aspect (1) of the abstract), the manuscript asserts that the second eigenvalue is simple except at the isolated critical height ĥ where multiplicity equals 2, and that the symmetry type of a continuous branch of u switches exactly from antisymmetric (h < ĥ) to symmetric (h > ĥ). Because a two-dimensional eigenspace at ĥ admits linear combinations with mixed symmetry, the argument must explicitly rule out additional crossings of the symmetric and antisymmetric branches elsewhere in the interval and confirm that the ordering of the first two eigenvalues remains consistent with the claimed simplicity for h ≠ ĥ. This justification is load-bearing for the symmetry classification.

    Authors: We agree that an explicit verification of the absence of additional crossings is essential. In the revised manuscript we will insert a new lemma that compares the first eigenvalues of the symmetric and antisymmetric subspaces as functions of h. Using the variational characterization and the monotonicity of the Rayleigh quotient with respect to the height parameter, we prove that these two curves intersect at most once. Combined with the already-established fact that they do intersect at ĥ (where multiplicity is two), this shows there are no further crossings. We will also add a short argument confirming that the second eigenvalue of the full problem coincides with the smaller of the two subspace eigenvalues and is therefore simple for h ≠ ĥ. These additions will be placed immediately after the continuity-method setup in Section 3. revision: yes

  2. Referee: The same issue arises for kite domains (aspect (3)): the statements that u is symmetric for h < h0 and antisymmetric for h > h1, with possible multiplicity-2 points at h0 and h1, require a detailed verification that the second eigenvalue stays simple away from those isolated values and that no further intersections occur between the even and odd branches in the parameter ranges 0 < a < 2. Without this, the claimed switch in symmetry type cannot be guaranteed for the full interval.

    Authors: We acknowledge the need for a more detailed crossing analysis for the kite family. In the revision we will add a proposition (in the section treating kites) that establishes the following: for each fixed a ∈ (0,2), the even and odd eigenvalues are continuous and strictly monotone in h; they intersect at most twice; and the ordering reverses exactly once between h0 and h1. The proof relies on the same eigenvalue-comparison technique used for trapezoids, together with an explicit computation of the second eigenvalue on the degenerate rectangle limit (h→0) and on the degenerate triangle limit (h→∞). These facts will guarantee that the second eigenvalue remains simple except at the isolated points h0 and h1, and that the symmetry type of the second eigenfunction switches only at those points. revision: yes

  3. Referee: The location of non-vertex critical points on the closure (claimed for both trapezoids and kites) relies on the symmetry classification together with maximum-principle or nodal-domain arguments. These steps must be shown to remain valid at the multiplicity-2 points and for all parameter values in the stated ranges; any dependence on the eigenfunction being strictly symmetric or antisymmetric needs explicit justification when multiplicity occurs.

    Authors: We will clarify this point in the revised text. At the multiplicity-two values (ĥ for trapezoids, h0 and h1 for kites) the eigenspace is spanned by one symmetric and one antisymmetric eigenfunction. Any eigenfunction in this space is a linear combination; its critical points on the boundary and interior can still be located by the same nodal-domain counting and boundary-point lemma that we already apply to the pure symmetric and antisymmetric cases, because the zero set of any nontrivial combination is contained in the union of the zero sets of the two basis functions. We will add a short paragraph after the critical-point theorems stating that the location statements therefore hold uniformly, including at the multiplicity-two parameters, and that the arguments do not require strict symmetry or antisymmetry. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard independent methods

full rationale

The paper derives symmetry properties of the second Neumann eigenfunction via symmetry decomposition of the eigenfunction space, direct eigenvalue comparisons between symmetric and antisymmetric subspaces, and application of the continuity method in the height parameter. These steps rely on variational characterizations and deformation arguments that are self-contained within the PDE setting and do not reduce by construction to the target symmetry statements or critical-height values. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the chain; the multiplicity-2 points are handled by isolating the transition and tracking branches separately, preserving logical independence from the claimed outcomes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard spectral theory of the Neumann Laplacian on bounded Lipschitz domains together with comparison and continuity arguments that are standard in the field.

axioms (1)
  • standard math The Neumann Laplacian on a bounded polygonal domain possesses a discrete spectrum of eigenvalues with corresponding eigenfunctions that are C^∞ in the interior.
    Invoked implicitly when discussing the second eigenfunction and its critical points.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The hot spots conjecture on Gaussian spaces

    math.SP 2024-12 unverdicted novelty 6.0

    Proves hot spots conjecture holds for lip domains and n-symmetric domains in Gaussian spaces via Hodge Laplacian variational methods.

Reference graph

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