Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces. Appendix to a note of A. Ros (Q1099420)

We use the classical Alexandrov reflection technique to prove that compact embedded hypersurfaces without boundary (in \({\mathbb{R}}^{n+1})\) must be Euclidean spheres, if they have constant intermediate curvature \(H_ r\) (the rth symmetric function of the n principal curvatures, \(1\leq r\leq n)\). The proof follows from a simple observation related to work by \textit{R. Ros} (see the preceding review), the curvature function \(H_ r\) is necessarily ``elliptic'' on any candidate surface, so the a priori convexity assumptions made in previous works are unnecessary.