Proper Lk-biharmonic Hypersurfaces in The Euclidean Sphere with Two Principal Curvatures

Document Type : Research Paper

Authors

Dept. of Math, Rafsanjan University of Vali-e-Asr, Iran

10.22103/jmmrc.2021.15736.1116

Abstract

In this paper we classify proper $L_k$-biharmonic hypersurfaces $ M $, in the unit Euclidean sphere which has two principal curvatures and we show that they are open pieces of standard products of spheres. Also we study proper $L_k$-biharmonic compact hypersurfaces $ M $ with respect to $tr(S^2\circ P_k)$ and $ H_k $ where $ S $ is the shape operator, $ P_k $ is the Newton transformation and $ H_k $ is the $ k $-th mean curvature of $ M $, and by definiteness's assumption of $ P_k $, we show that $ H_{k+1} $ is constant.

Keywords