On the rigidity of self-shrinkers of the \(r\)-mean curvature flow (Q6543118)

The authors consider self-similar solutions of the \(r\)-mean curvature flow in $(n+ 1)$-dimensional Euclidean space. \N\NGeometric flows have been an active research field in recent decades. \textit{G. Huisken} and \textit{A. Polden} [Lect. Notes Math. 1713, 45--84 (1999; Zbl 0942.35047)] presented the evolution of hypersurfaces in Riemannian manifolds moving in the direction of their normal vector field with speed given by suitable functions of the principal curvatures. A special family of functions is the \(r\)th-elementary symmetric polynomials, and the geometric flow associated with them has nice properties. In this paper, the authors consider this flow for complete, not necessarily compact, hypersurfaces in \({\mathbb R}^{n+1}\) that are weakly convex, and also satisfy other natural geometric constraints. By employing some general maximum principles, some self-similar solutions of the \(r\)-mean curvature flow under some suitable geometric constraints are characterised.