Hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$ with constant sectional curvature

Abstract

In this paper, we classify the hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$ 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 $\frac{1}{2}$ 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 ${\mathbb{S}}^2\times {\mathbb{S}}^2$ with $C=0$, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous ``sinh-Gordon equation'' $$(\frac{{\partial }^2}{\partial u^2}+\frac{{\partial }^2}{\partial v^2})h=-\frac{1}{\sqrt{2}}\sinh (\sqrt{2}h).$$ As a byproduct, we give a complete classification of the hypersurfaces of ${\mathbb{S}}^2\times {\mathbb{S}}^2$ with constant mean curvature and constant product angle function $C$.

24 pages