Integral-Einstein hypersurfaces in spheres

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 is the main object of the Chern conjecture: such hypersurfaces are isoparametric. 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 ( 1 n ) × S n−1 ( 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 ≤ 2 principal curvatures. In particular, all the minimal isoparametric hypersurfaces with g ≥ 3 principal curvatures are IE hypersurfaces. As applications, we also obtain some spherical Bernstein theorems.

A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces

“…More related results can be found in the surveys by Scherfner-Weiss [49], Scherfner-Weiss-Yau [50] and Ge-Tang [29]. Very recently, we [28] gave a characterization to the condition of Chern Conjecture 1.2, a Takahashi-type theorem, i.e., An immersed hypersurface M n in S n+1 is minimal and has constant scalar curvature if and only if ∆ν = λν for some constant λ, where ν is a unit normal vector field of M n .…”

“…In this paper, without assuming constant scalar curvature, we are interested in whether there is a universal lower bound for the mean value of S on non-totally geodesic minimal hypersurfaces. Inspired by the preceding paper [28], we answer this question affirmatively for embedded hypersurfaces.…”

“…Let x : M n S n+1 ⊂ R n+2 be a closed immersed hypersurface in the unit sphere S n+1 and ν(x) denote the unit normal vector field, ∇ and ∇ be the Levi-Civita connections on M n and S n+1 , respectively. Following [28], for any fixed unit vector a ∈ S n+1 , we consider the height functions on M n :…”

The isoperimetric inequalities of minimal hypersurfaces in spheres

“…For instance, the well known Takahashi's theorem [19] stated that M n is minimal if and only if there exists a constant λ such that ∆ϕ a = −λϕ a for all a ∈ S n+1 . Analogously, Ge and Li [9] gave a Takahashitype theorem, i.e., an immersed hypersurface M n in S n+1 is minimal and has constant scalar curvature (CSC) if and only if ∆ν = λν for some constant λ. The condition of the constant scalar curvature (CSC) is linked to the famous Chern Conjecture (cf.…”

A note on the Chern Conjecture in dimension four

“…Very recently, using the height functions of the normal vector field (cf. [17,24]), Ge-Li [16] proved that there is a positive constant δ(n) > 0 depending only on n such that on any closed embedded, non-totally geodesic, minimal hypersurface…”