Integral-Einstein hypersurfaces in spheres
Jianquan Ge, Fagui Li
TL;DR
The work introduces Integral-Einstein (IE) submanifolds as an intrinsic–extrinsic generalization of Einstein manifolds and develops a Takahashi-type framework for hypersurfaces in the unit sphere. Using height functions $\varphi_a$ and $\psi_a$, the authors establish criteria for constant mean curvature and constant squared norm of the second fundamental form, and provide a precise integral characterization of IE hypersurfaces, distinguishing totally geodesic, Clifford torus, and IE minimal CSC cases. They leverage Reilly's formula and the Cheng–Yau operator to prove the IE criteria and demonstrate that minimal isoparametric hypersurfaces with $g\ge 3$ are IE, while the Clifford torus is not IE, with sharp volume-based equalities linked to these classes. The paper then derives sharp $L^2$-norm inequalities for the height functions, leading to spherical Bernstein-type theorems that constrain how non-totally geodesic minimal hypersurfaces can sit near equators or within spherical zones, thereby connecting IE geometry to rigidity phenomena in spherical ambient spaces.
Abstract
Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres which are conjectured to be isoparametric in the Chern conjecture. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with $g\leq2$ principal curvatures. In particular, all minimal isoparametric hypersurfaces with $g\geq3$ principal curvatures are IE hypersurfaces. As applications, we obtain some spherical Bernstein theorems, including that any embedded closed minimal surface of genus no more than $\mathfrak{g}$ inside a tubular neighborhood of constant radius $r(\mathfrak{g})>0$ around an equator in $\mathbb{S}^3$ is an equator.