Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure

It is known that if the Gaussian curvature function along each meridian on a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ decreasing, then cut locus point $\theta^{-1} (0)$ empty or subarc opposite (\pi)$. Such called von Mangoldt's revolution. A generalized Mangoldt For example, m_0(r)^{2} d\theta^{2})$, where $m_{0}(x) = x/(1 x^{2})$, has same structure as above and in $r^{-1}((0, \infty))$ nonempty. Note not decreasing for this surface. In article, we give sufficient conditions to be Moreover, prove any with finite total $c$, there exists $c$ such monotone $[a, \infty)$ $a > 0$.