Rotational surfaces of prescribed Gauss curvature in $\mathbb{R}^3$

TL;DR

This paper investigates rotational surfaces in three-dimensional space with a prescribed Gauss curvature function, extending classical classifications, providing new examples, and analyzing their asymptotic and singular behaviors.

Contribution

It generalizes the classification of rotational surfaces to prescribed Gauss curvature functions, introduces new non-constant curvature examples, and studies asymptotic and singular solutions.

Findings

Generalized classification of rotational surfaces with prescribed Gauss curvature

Constructed examples not possible in constant curvature case

Analyzed asymptotic behavior of convex graphs and singular solutions

Abstract

We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the classification of rotational surfaces of constant Gauss curvature; exhibit examples that cannot exist in the constant Gauss curvature case; and analyze the asymptotic behavior of strictly convex graphs. We also prove the existence of singular radial solutions intersecting orthogonally the axis of rotation.