Constant mean curvature hypersurfaces with two principal curvatures in a sphere

Abstract

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by ${\phi_{ij}}$ the trace free part of the second fundamental form of M ⁿ, and Φ be the square of the length of ${\phi_{ij}}$ . We obtain two integral formulas by using Φ and the polynomial ${P_{H,m}(x)=x^{2}+ \frac{n(n-2m)}{\sqrt{nm(n-m)}}H x -n(1+H^{2})}$ . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either ${\Phi=B_{H,m}}$ or ${\Phi=B_{H,n-m}}$ . In particular, M ⁿ is the hypersurface ${S^{n-m}(r)\times S^{m}(\sqrt{1-r^{2}})}$ .