Rotationally invariant constant Gauss curvature surfaces in Berger\n spheres

2019/12/05 by Francisco Torralbo, Torralbo, Francisco, Joeri Van der Veken +1
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1912.02681

Abstract

We give a full classification of complete rotationally invariant surfaces\nwith constant Gauss curvature in Berger spheres: they are either Clifford tori,\nwhich are flat, or spheres of Gauss curvature K \≥ K₀ for a positive\nconstant K₀, which we determine explicitly and depends on the geometry of\nthe ambient Berger sphere. For values of K₀ \≤ K \≤ K_P, for a specific\nconstant K_P, it was not known until now whether complete constant Gauss\ncurvature K surfaces existed in Berger spheres, so our classification\nprovides the first examples. For K > K_P, we prove that the rotationally\ninvariant spheres from our classification are the only topological spheres with\nconstant Gauss curvature in Berger spheres.\n

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