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

Abstract

It was proven in [24] that if the Gaussian curvature function along each meridian on a surface of revolution (R 2 , dr 2 + m(r) 2 dθ 2 ) is decreasing, then the cut locus of each point of θ −1 (0) is empty or a subarc of the opposite meridian θ −1 (π). Such a surface is called a von Mangoldt's surface of revolution in [24] (see also [17]). A surface of revolution (R 2 , dr 2 + m(r) 2 dθ 2 ) is called a generalized von Mangoldt surface of revolution if the cut locus of each point of θ −1 (0) is empty or a subarc of the opposite meridian θ −1 (π).For example, the surface of revolution (R 2 , dr 2 + m 0 (r) 2 dθ 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, ∞)) 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 (R 2 , dr 2 + m(r) 2 dθ 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, ∞) for any a > 0.