Hypersurfaces with constant higher order mean curvature in Euclidean space (Q292056)

The authors consider complete oriented hypersurfaces \(M\) in \(\mathbb R^{n+1}\), \(n \geq 3\), with constant mean curvature \(H\) and only two distinct principal curvatures, one of them of multiplicity one. For such hypersurfaces, the first author and \textit{S. C. García-Martínez} provide the upper bound \(n^2H^2\) for the infimum of the squared norm of the shape operator [ibid. 156, 31--47 (2012; Zbl 1232.53046)]. In the present paper, this is extended to hypersurfaces with constant positive \(r\)-th mean curvature \(H_r\): The number \((n-1)(nH_r/(n-r))^{2/r}\) is an upper bound for the infimum as well as a lower bound for the supremum of the squared norm of the shape operator over \(M\). Each bound is sharp if and only if \(M\) is isometric to a suitable cylinder over \(\mathbb S^{n-1}\). This paper uses techniques different from related recent publications by different authors. Proofs are based on the Principal Curvature Theorem by \textit{B. Smyth} and \textit{F. Xavier} [Invent. Math. 90, 443--450 (1987; Zbl 0642.53007)].