2026/07/20 by Haiyun Deng, Xuyong Jiang, Xiaoping Yang
Mathematics · #math.AP
In this paper, we investigate critical points of second Neumann eigenfunctions on convex domains in the two-dimensional space forms. We approach this problem from three complementary perspectives: spectral and geometric conditions; explicit quantitative location restrictions; the hot spots constant. Precisely, for the spectral and geometric conditions, we prove that if a convex domain Ω is contained in the hemisphere and satisfies μ2(Ω)≤ 2, then its second Neumann eigenfunction has no interior critical points. Beyond this, we establish a unified diameter-based criterion μ2(Ω)D2≤ j1,12 ensuring the absence of interior critical points in \mathbbS2 and ℍ2. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain's diameter in \mathbbS2 and ℍ2. Finally, we study the hot spots constant \mathfrakC(Ω) on convex domains using purely analytical methods. We refine the known Euclidean upper bound of \mathfrakC(Ω) to 2.4828, and obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms for the first time. Our proofs combine the properties of Bessel and Legendre functions, estimation of eigenvalues and Green formulas. Our results quantitatively measure ``how wrong'' the hot spots conjecture can be.