Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms
Abstract
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 𝕊2 and ℍ2. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if μ2(Ω)D2≤ j1,12, 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 𝕊2 and ℍ2. Finally, we develop an analytic approach to study the \emph{hot spots constant} C(Ω) on convex domains. For planar convex domains, we improve the Euclidean upper bound to C(Ω)<1.48. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, C(Ω) < 4 for Ω⊂𝕊2 contained in a hemisphere; C(Ω) < 11.2 for Ω⊂ℍ2. 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.
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BibTeX
@misc{deng2026quantitative,
title = {{Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms}},
author = {Haiyun Deng and Xuyong Jiang and Xiaoping Yang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.17882},
eprint = {2607.17882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17882},
}
RIS
TY - UNPB
TI - Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms
AU - Haiyun Deng
AU - Xuyong Jiang
AU - Xiaoping Yang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2607.17882
UR - https://arxiv.org/abs/2607.17882
AB - 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 𝕊^2 and 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≤ 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 𝕊^2 and ℍ^2. Finally, we develop an analytic approach to study the \emph{hot spots constant} C(Ω) on convex domains. For planar convex domains, we improve the Euclidean upper bound to C(Ω)<1.48. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, C(Ω) < 4 for Ω⊂𝕊^2 contained in a hemisphere; C(Ω) < 11.2 for Ω⊂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.
ER -