Rotationally invariant constant Gauss curvature surfaces in Berger spheres

TL;DR

This paper classifies all complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres, identifying new examples and establishing uniqueness results for certain curvature ranges.

Contribution

It provides a complete classification of such surfaces, explicitly determines the curvature bounds, and introduces the first known examples for specific curvature values.

Findings

Clifford tori are flat solutions.

Existence of spheres with curvature $K eq 0$ depending on ambient geometry.

Uniqueness of rotationally invariant spheres for $K > K_P$.

Abstract

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant $K_0$, which we determine explicitly and depends on the geometry of the ambient Berger sphere. For values of $K_0 \leq K \leq K_P$, for a specific constant $K_P$, it was not known until now whether complete constant Gauss curvature $K$ surfaces existed in Berger spheres, so our classification provides the first examples. For $K > K_P$, we prove that the rotationally invariant spheres from our classification are the only topological spheres with constant Gauss curvature in Berger spheres.