Critical points of second Neumann eigenfunctions on some convex domains in two-dimensional space forms

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 $\Omega$ is contained in the hemisphere and satisfies $\mu_{2}(\Omega)\leq 2$, then its second Neumann eigenfunction has no interior critical points. Beyond this, we establish a unified diameter-based criterion $\mu_2(\Omega)D^2\leq j_{1,1}^2$ ensuring the absence of interior critical points in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain's diameter in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we study the hot spots constant $\mathfrak{C}(\Omega)$ on convex domains using purely analytical methods. We refine the known Euclidean upper bound of $\mathfrak{C}(\Omega)$ 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.