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.

A maximum principle for hypersurfaces with constant scalar curvature and applications

In this paper we study the rigidity of complete hypersurfaces with constant scalar curvature in Riemannian space forms. Under an appropriate constraint on Φ, the traceless part of its second fundamental form, we prove that either the hypersurface is totally umbilical or it holds a sharp estimate for the supremum of the norm of Φ, with equality if and only if the hypersurface is isoparametric with two distinct principal curvatures. Moreover, we also construct complete non-isoparametric rotational examples which show that our constraint on Φ is sharp and necessary.

Hypersurfaces with constant scalar curvature in space forms

In this paper we deal with complete linear Weingarten hypersurfaces immersed into Riemannian space forms. Assuming an Okumura type inequality on the total umbilicity tensor of such hypersurfaces, we prove that either the hypersurface is totally umbilical or it holds an estimate for the norm of the total umbilicity tensor, which is also shown be sharp in the sense that the product of space forms realize them. Our approach is based on a version of the Omori–Yau maximum principle for a suitable Cheng–Yau type operator.

Complete weingarten hypersurfaces satisfying an okumura type inequality

Let M be an n(n ‚ 3)-dimensional complete connected hyper- surface in a unit sphere S n+1 (1). In this paper, we show that (1) if M has non-zero mean curvature and constant scalar curvature n(ni1)r and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S 1 ( p 1 i c 2 ) £ S ni1 (c), c 2 = ni2 nr if r ‚ ni2 ni1

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Hypersurfaces with constant scalar or mean curvature in a unit sphere

We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Melendez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.

A gap theorem for constant scalar curvature hypersurfaces

In this paper, we give a characterization of Clifford tori and in a unit sphere S n +1 (1). Our results extend the results due to Cheng and Yau [ 4 ], and Wang and Xia [ 11 ].

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Guoxin Wei

Papers

Rigidity theorems for hypersurfaces with constant scalar curvature in a unit sphere