Unstable nonvertex critical points of second Neumann eigenfunctions on convex quadrilaterals: unconditional existence and consequences for the deformation approach to the hot spots conjecture

For a second Neumann eigenfunction on a convex polygon, Judge and Mondal (Comm. PDE 2022) proved conditionally that if no convex quadrilateral has an interior critical point, then some convex quadrilateral has a nonvertex critical point that is unstable under domain perturbation. The problem is to determine the unconditional status of this instability phenomenon: do unstable nonvertex critical points exist on convex quadrilaterals (the authors conjecture they do not occur on triangles), and does the answer thereby settle whether the phenomenon is a genuine unconditional obstruction to the deformation (continuity) strategy for proving the strong hot spots conjecture on polygonal domains.

Background

Let $\Omega$ be a bounded planar domain and let $-\Delta$ act on functions satisfying a Neumann boundary condition, that is, the normal derivative of the function vanishes on $\partial\Omega$ (an insulated body). The eigenfunctions associated with the smallest nonzero eigenvalue $\mu_2$ — the second Neumann eigenfunctions — describe the slowest-decaying spatial temperature mode. The hot spots conjecture of Rauch (1974) asserts that every second Neumann eigenfunction attains its maximum and minimum only on the boundary of $\Omega$. The conjecture is false for some planar domains with holes (Burdzy–Werner 1999) but is believed true for convex and for simply connected planar domains; it is known for 'lip' domains — planar domains between the graphs of two Lipschitz functions of Lipschitz constant 1 (Atar–Burdzy 2004, with a non-probabilistic proof by Rohleder 2021) — and, in a landmark result, for all Euclidean triangles: Judge and Mondal (Ann. of Math. 2020, with a 2022 erratum) proved that a second Neumann eigenfunction on a triangle has no interior critical points, and Chen–Gui–Yao (Invent. Math. 2026) further proved that it has at most one nonvertex critical point and is monotone in some fixed direction. Since every planar domain is approximable by polygons, the strong form of the conjecture — every second Neumann eigenfunction on every polygon has no interior maximum — would imply the weak form (some second Neumann eigenfunction has no interior maximum) for all planar domains.

A critical point is a point where $\nabla u=0$. At a vertex of a Neumann polygon the gradient vanishes automatically, so the interesting objects are nonvertex critical points, lying in the interior or on the sides. Judge and Mondal's approach to triangles and polygons is a deformation (continuity) method: one deforms the polygon continuously to a terminal domain whose critical-point structure is known, and tracks the critical points along the path; the method succeeds when the critical points are stable. Formally, a nonvertex critical point $p$ of a second Neumann eigenfunction $u$ on a polygon $Q$ is stable under perturbation if whenever polygons $Q_n$ converge to $Q$ and second Neumann eigenfunctions $u_n$ on $Q_n$ converge to $u$, each $u_n$ has a critical point $p_n$ converging to $p$. For triangles the method was carried through completely; for Lip-1 polygons with no orthogonal sides, Theorem 1.2 of the source paper shows the second Neumann eigenvalue is simple and the only critical points are the two acute vertices.

Beyond these classes the source paper (Judge–Mondal, Comm. PDE 47 (2022) 1559–1590) encounters a new phenomenon. Its Theorem 1.1 states: if each convex quadrilateral has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral $Q$, a second Neumann eigenfunction $u$ on $Q$, and a nonvertex critical point $p$ of $u$ that is not stable under perturbation. The proof mechanism is 'blocking'. Near a vertex of angle $\beta$ an eigenfunction has a Fourier–Bessel expansion

$$u\left(re^{i\theta}\right)=\sum_{n=0}^{\infty}c_n\,J_{n\pi/\beta}\left(\sqrt{\mu}\,r\right)\cos\left(\frac{n\pi\theta}{\beta}\right),$$

where $J_\nu$ is the Bessel function of the first kind; the first two coefficients $c_0,c_1$ control the local behavior. Each isolated critical point carries a variant of the Poincaré–Hopf index (a topological invariant obtained by winding $\nabla u$ around the point, extended in the source to vertex critical points), whose total is preserved under perturbation. An index $-1$ critical point lying on a side cannot cross an obtuse vertex of angle in $(\pi/2,\pi)$, because passage would force $c_1=0$ at that vertex, making the vertex an index $+1$ point and abruptly changing the total index. The paper's Proposition 8.1 turns this blocking phenomenon into a sufficient criterion: along a continuous family of quadrilaterals with one obtuse vertex whose adjacent sides host the moving index $-1$ critical point (plus nondegeneracy-type hypotheses), an unstable critical point must occur. Theorem 9.5 then constructs such families by 'breaking' an acute triangle of a suitable open set $N$ along the side containing its index $-1$ critical point — but only under the hypothesis that the resulting quadrilaterals have no interior critical points, which is exactly what is unproved. The authors conjecture that instability does not occur for triangles. Recent work of Deng–Gui–Jiang–Yang–Yao–Zou (2026) fully characterizes the nonvertex critical points of second Neumann eigenfunctions on symmetric quadrilaterals (isosceles trapezoids, parallelograms — where there are none — rhombi, kites) and proves hot spots for these families, but does not study stability under perturbation; for general convex quadrilaterals and polygons with more sides both hot spots and the critical-point structure remain open, so the unconditional status of the instability phenomenon is unknown.

Problem Statement

Let $Q$ be a convex polygon and $u$ a second Neumann eigenfunction on $Q$. A nonvertex critical point $p$ of $u$ is called stable under perturbation if, whenever polygons $Q_n$ converge to $Q$ and second Neumann eigenfunctions $u_n$ on $Q_n$ converge to $u$, each $u_n$ has a critical point $p_n$ converging to $p$; $p$ is unstable if it is not stable. Convergence is understood in the sense of the source paper (Judge–Mondal, Comm. PDE 2022, Section 4): $Q_n$ converges to $Q$ when there are homeomorphisms $\phi_n\colon Q\to Q_n$ that are $C^2$ diffeomorphisms away from the vertices, with $\phi_n\to\mathrm{id}$ uniformly in $C^2$ on compact subsets excluding the vertices, and $u_n$ converges to $u$ when $u_n\circ\phi_n$ converges to $u$.

Stated in the source's own terms, the problem is the unconditional status of Theorem 1.1 of Judge–Mondal (2022): does there exist a convex quadrilateral $Q$, a second Neumann eigenfunction $u$ on $Q$, and a nonvertex critical point $p$ of $u$ that is unstable? Theorem 1.1 establishes this only under the unproved hypothesis that no convex quadrilateral has a second Neumann eigenfunction with an interior critical point; the unconditional status — existence, or nonexistence on all convex quadrilaterals — is unknown. Determine what the existence, or the nonexistence, of such a critical point decides about the deformation (continuity) strategy for the strong hot spots conjecture on polygonal domains, in particular whether the instability phenomenon is a genuine unconditional obstruction to that strategy. The authors conjecture that the analogous statement fails for triangles, i.e. that instability does not occur for triangles.

The source frames this question through its deformation (continuity) strategy for the strong hot spots conjecture, observing that the strategy is likely to be more difficult to implement if the polygon has more sides: an unconditional existence result or construction would show that unstable critical points form a genuine obstruction to the method as formulated for polygons with four or more sides, while a nonexistence or rigidity result would remove the obstruction identified in the source paper.

References

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