Linear Weingarten Hypersurfaces in a Unit Sphere

Abstract

Abstract. In this paper, we have considered linear Weingarten hypersurfaces in a sphere and obtained some rigidity theorems. The purpose of this paper is to give some extension of the results due to Cheng-Yau [3] and Li [7].

Linear Weingarten Hypersurfaces in Riemannian Space Forms

“…As a natural generalization of hypersurface with constant scalar curvature or with constant mean curvature, linear Weingarten hypersurface has been studied in many places ( [5,8,9,11,13,15,16]). Recall that a hypersurface in a Riemannian space form is said to be linear Weingarten if its normalized scalar curvature R and mean curvature H satisfy R = aH + b for some constants a, b ∈ R. In [13], Li, Suh and Wei proved the first rigidity result for linear Weingarten hypersurface in S n+1 (1) under the assumption that the hypersurface is compact. Later, Shu [15] extended this rigidity result to the case of complete linear Weingarten hypersurface with two distinct principle curvatures in real space forms.…”

“…In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form M n+1 (c) (c = 1, 0, −1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in S n+1 (1) with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface. …”

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

“…In particular, they extended the main theorem of [7] for the context of Q n+1 c . Meanwhile, Li, Suh and Wei [12] considered the so-called linear Weingarten hypersurfaces immersed in S n+1 , that is, hypersurfaces of S n+1 whose mean curvature H and normalized scalar curvature R satisfy R = aH +b, for some a, b ∈ R. In this setting, they showed that if M n is a compact linear Weingarten hypersurface with nonnegative sectional curvature immersed in S n+1 such that R = aH + b with (n − 1)a 2 + 4n(b − 1) ≥ 0, then M n is either totally umbilical or isometric to a Clifford torus. More recently, the first, second and fourth authors [6] obtained another characterization result concerning complete linear Weingarten hypersurfaces immersed in Q n+1 c…”

“…To obtain the second lemma, we can argue as in the proof of either Lemma 2.1 of [12] or Lemma 3.2 of [6]. In what follows, as before, R = 1 n(n−1) i,jR ijij .…”

Linear Weingarten Hypersurfaces in a Real Space Form

“…Suh and G. Wei [7] introduced the so called linear Weingarten hypersurface in a unit sphere S n+1 (1). We can generalize it to a real space form M n+1 (c), that is, a hypersurface in a real space form M n+1 (c) is called a linear Weingarten hypersurface if the scalar curvature R and the mean curvature H satisfy the linear relation αR + βH + γ = 0, where α, β and γ are constants such that α 2 + β 2 = 0.…”