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 ${\mathbb{S}}^2$ and ${\mathbb{H}}^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if ${\mu }_2(\Omega )D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $\Omega$ 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} $\mathfrak{C}(\Omega )$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(\Omega )<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(\Omega )<4$ for $\Omega \subset {\mathbb{S}}^2$ contained in a hemisphere; $\mathfrak{C}(\Omega )<11.2$ for $\Omega \subset {\mathbb{H}}^2$. 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.

40pages, 1 figure