2023/02/01 by Haizhong Li, Luc Vrancken, Li, Haizhong +5
Mathematics · Physics and Astronomy · #53B25 #53C42 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2302.00466
openalex publication_date 2023/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we classify the hypersurfaces of \mathbbS2×\mathbbS2 with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be (1)/(2) and the product angle function C defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in \mathbbS2×\mathbbS2 with C=0, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous ``sinh-Gordon equation'' ((∂2)/(∂ u2)+(∂2)/(∂ v2))h =-\tfrac1√(2)\sinh(√(2)h). As a byproduct, we give a complete classification of the hypersurfaces of \mathbbS2×\mathbbS2 with constant mean curvature and constant product angle function C.