Xuyong Jiang | Searcharxiv

arXiv2026

Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites

In this paper, we determine the symmetry properties and non-vertex critical points of the second Neumann eigenfunction $u$, as well as the multiplicity of the corresponding eigenvalue, on isosceles trapezoids and kites. By exploiting reflection symmetry, we reduce the problem to a comparison between the second Neumann eigenvalue and the first mixed Dirichlet--Neumann eigenvalue on the half-domain. More precisely, for isosceles trapezoids with base angle $α\leq \fracπ{3}$, the second eigenfunction is antisymmetric. If $\fracπ{3}<α<\fracπ{2}$, there exists a critical height $\hat{h}(α)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(α)$ and symmetric when $h>\hat{h}(α)$, while at $h=\hat{h}(α)$ the second Neumann eigenvalue has multiplicity two. For a convex kite $P_1P_2P_3P_4$, where $P_1=(0,0)$, $P_2=(a,-h)$, $P_3=(1,0)$, and $P_4=(a,h)$, an analogous result holds: there exists a critical height $\tilde{h}(a)$ such that $u$ is symmetric with respect to the $x$-axis when $h<\tilde{h}(a)$ and antisymmetric with respect to the $x$-axis when $h>\tilde{h}(a)$, while at $h=\tilde{h}(a)$ the second Neumann eigenvalue has multiplicity two. In all cases where the second Neumann eigenvalue is simple, we determine all non-vertex critical points of the corresponding eigenfunction, thereby verifying the hot spots conjecture.

math.AP↗

arXiv2026

Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $μ_2(Ω)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $Ω$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the \emph{hot spots constant} $\mathfrak{C}(Ω)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(Ω)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(Ω) < 4$ for $Ω\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(Ω) < 11.2$ for $Ω\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure ``how wrong'' the \emph{hot spots conjecture} can be.

math.AP↗