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 (ℝ², 𝑑𝑟² + 𝑚(𝑟)² 𝑑𝜃²) 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. A surface of revolution (ℝ², 𝑑𝑟² + 𝑚(𝑟)² 𝑑𝜃²) 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 (ℝ², 𝑑𝑟² + 𝑚₀(𝑟)² 𝑑𝜃²), where 𝑚₀(𝑥) = 𝑥/(1 + 𝑥²), 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 (ℝ², 𝑑𝑟² + 𝑚(𝑟)² 𝑑𝜃²) 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.