A maximum principle for hypersurfaces with constant scalar curvature and applications

Abstract

In this article, we establish a weak maximum principle for complete hypersurfaces with constant scalar curvature into Riemannian space forms, and give some applications to estimate the norm of the traceless part of its second fundamental form.

Linear Weingarten Hypersurfaces in Riemannian Space Forms

“…Thus Σ n is an isoparametric hypersurface. Then, by the classical result in [4], we deduce that Σ n satisfies the three following standard product embeddings:…”

“…In this paper, we firstly obtain a weak maximum principle (see Lemma 2.9 below) for the operator L defined by (2.4) which can be regarded as a generalization of Corollary 10 in [4]. Then, using this maximum principle, we can extend the rigidity results in [13] and [15] to the case of complete linear Weingarten hypersurface without the assumption of two distinct principle curvatures in M n+1 (c).…”

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

“…In [3], by applying a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti [15], Alías and , deriving a sharp estimate for the infimum of R. Afterwards, Alías, García-Martínez and Rigoli [4] obtained another suitable weak maximum principle for complete hypersurfaces with constant scalar curvature in Q n+1 c…”

Linear Weingarten submanifolds in unit sphere

“…Riemannian space form is said to be linear W eingarten if its (normalized) scalar curvature R and its mean curvature H are related by R = aH + b for some constants a, b ∈ R. In [13], Li et al proved the first rigidity result for linear Weingarten hypersurfaces under the assumption that the hypersurface is compact in S n+1 (1). Later, we generalized it to the case of complete hypersurfaces with the aid of a weak maximum principle in [7].…”

“…In this paper, we go on studying this kind of linear Weingarten submanifolds and obtain the following result: Theorem 1.1. Let Σ n (n ≥ 4) be a compact submanifold in unit sphere S n+p (1) with parallel normalized mean curvature vector and R = aH…”