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

Abstract

It is known that if the Gaussian curvature function along each meridian on a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is decreasing, then the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$. Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$.

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