A Sharp Diameter-Dependent Lower Bound for the First Nonzero Neumann Eigenvalue of Geodesic Triangles in Space Forms

We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When $K>0$ and $D=\pi \mathrm{/}(2\sqrt{K})$, equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most $\pi \mathrm{/}2$, and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter $\pi \mathrm{/}2$.