LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

Abstract

Abstract. In this note, we generalize the weak maximum principle in[4] to the case of complete linear Weingarten hypersurface in Riemannianspace form M n+1 (c) (c = 1,0,−1), and apply it to estimate the normof the total umbilicity tensor. Furthermore, we will study the linearWeingarten hypersurface in S n+1 (1) with the aid of this weak maximumprinciple and extend the rigidity results in Li, Suh, Wei [13] and Shu [15]to the case of complete hypersurface. 1. IntroductionIt is well known that many rigidity results have been obtained for hyper-surfaces in spheres ([1, 6, 10, 17]) and in space form ([2, 3, 12]) with constantscalar curvature or with constant mean curvature. As a natural generaliza-tion of hypersurface with constant scalar curvature or with constant meancurvature, linear Weingarten hypersurface has been studied in many places([5, 8, 9, 11, 13, 15, 16]). Recall that a hypersurface in a Riemannian spaceform is said to be linear Weingarten if its normalized scalar curvature R andmean 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 hypersurfacein S