On closed hypersurfaces of constant scalar curvatures and mean curvatures inSⁿ⁺¹

Abstract

We consider in this note the following question: given a closed Riemann n -manifold of constant scalar curvature, how can it be minimally immersed in the round (n + 1)-sphere? Our main result states that the immersion has to be isoparametric if the number of its distinct principal curvatures is three identically. This provides another piece of supporting evidence to a conjecture of Chern.

M�bius hypersurfaces in SS+1with three distinct principal curvatures

“…. , m 1 Using (4.5)-(4.7), (4.10), (4.16) and (4.17), by a similar calculation as in case (1), we have…”

“…By the argument of Chang [1], we know that m 1 , m 2 and m 3 are constant integers. Using indices convention: 1 ≤ i, j, k, .…”

On the Local Version of the Chern Conjecture: CMC Hypersurfaces with Constant Scalar Curvature in S n+1

“…Let M be an n-dimensional hypersurface in a unit sphere S n+1 (1). We choose a local orthonormal frame field {e 1 , .…”

“…. , ω n+1 denote the dual co-frame field in S n+1 (1). We use the following convention for the indices: A, B, C, D range from 1 to n + 1 and i, j, k from 1 to n. The structure equations of S n+1 (1) as a hypersurface of the Euclidean space R n+2 are given by…”

Closed hypersurfaces of S4(1) with constant mean curvature and zero Gauß–Kronecker curvature

“…Defining the functions µ i := λ i − H , we have that i µ i = 0. The following classical formulas for the Laplacians of S and f 3 are well known and can be found in many papers such as [5][6][7][8][9]:…”