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

TL;DR

This paper characterizes generalized von Mangoldt surfaces of revolution, extending classical results to surfaces with non-monotonic Gaussian curvature, and shows the existence of such surfaces with prescribed total curvature.

Contribution

It provides sufficient conditions for a surface of revolution to be a generalized von Mangoldt surface and constructs examples with non-monotonic curvature sharing the same total curvature.

Findings

Characterization of generalized von Mangoldt surfaces.

Existence of non-monotonic curvature surfaces with given total curvature.

Extension of cut locus structure understanding.

Abstract

It was known that if the Gaussian curvature function along each meridian on a surface of revolution $(R^2, dr^2+m(r)^2d\theta^2)$ 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 $(R^2, dr^2+m(r)^2d\theta^2)$ 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 $(R^2, dr^2+m_0(r)^2d\theta^2),$ 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 $(R^2, dr^2+m(r)^2d\theta^2)$ 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 $a>0.$