2022/03/11 by Hilário Alencar, Alencar, Hilário, Gregório Silva Neto +1
Mathematics · Physics and Astronomy · #30A10 #30F10 #53C21 #58J05 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 53C42 #Secondary 30F30
paper · pdf · doi:10.48550/arxiv.2203.06206
openalex publication_date 2022/03/11 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere,\nwith constant mean curvature in the Euclidean space are the round (geometrical)\nspheres. These results were generalized by S. S. Chern, and then by Eschenburg\nand Tribuzy, for surfaces, homeomorphic to the sphere, in Riemannian manifolds\nwith constant sectional curvature whose mean curvature function satisfies some\nbound on its differential. In this paper, using techniques partial differential\nequations in the complex plane which generalizes the notion of holomorphy, we\nextend these results for surfaces in a wide class of warped product manifolds,\nwhich includes, besides the classical space forms of constant sectional\ncurvature, the de Sitter-Schwarzschild manifolds and the Reissner-Nordstrom\nmanifolds, which are time slices of solutions of the Einstein field equations\nof the general relativity.\n