Constant mean curvature hypersurfaces with two principal curvatures in a sphere (Q836911)

The author obtains two integral formulas for compact hypersurfaces with constant mean curvature and two distinct principal curvatures with multiplicities \(n - m\) and \(m\), \(1\leq m \leq n-1\), immersed in the unit sphere \(S^{n+1}(1)\). The first formula involves \(\Phi\), the square length of the trace free part of the second fundamental form, and the polynomial \(P_{H,m}(x)=x^2+\frac{n(n-2m)}{\sqrt{nm(n-m)}}Hx -n(1+H^2)\), where \(H\) stands for the mean curvature. This polynomial is also used to establish the second integral formula, which generalizes another one previously obtained in: [\textit{L. J. Alías, S. C. D. Almeida} and \textit{A. Brasil, jun.}, An. Acad. Bras. Ciênc. 76, No.~3, 489--497 (2004; Zbl 1073.53071)]. By using the above two mentioned integral formulas, the author proves that a constant mean curvature compact oriented hypersurface \(M^n\), immersed in the unit sphere \(S^{n+1}(1)\) with two distinct principal curvatures of multiplicities \(n - m\) and \(m\), \(m\), \(1\leq m \leq n-1\), has to be the hypersurface \(S^{n-m}(r) \times S^m(\sqrt{1 - r^2})\).