A gap theorem for constant scalar curvature hypersurfaces
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A gap theorem for constant scalar curvature hypersurfaces Eudes L. de Lima1 · Henrique F. de Lima2 Received: 15 April 2020 / Accepted: 16 November 2020 © Universitat de Barcelona 2020
Abstract We obtain a sharp estimate to the norm of the traceless second fundamental form of com‑ plete hypersurfaces with constant scalar curvature immersed into a locally symmetric Rie‑ mannian 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 curva‑ tures. Our approach involves a suitable Okumura type inequality which was introduced by Meléndez (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. Keywords Locally symmetric Riemannian manifolds · Riemannian space forms · Complete hypersurfaces · Constant scalar curvature · Okumura type inequality Mathematics Subject Classification Primary 53C24 · 53C40 · 53C42
1 Introduction In 1977, Cheng and Yau [7] proved the following well known rigidity result concerning compact constant scalar curvature hypersurfaces immersed into a Riemannian manifold with constant sectional curvature which, in its original version, states:
Theorem (Theorem 2 of [7]) Let Σn be a compact hypersurface with nonnegative sec-
tional curvature immersed in a manifold with constant sectional curvature c. Suppose that the normalized scalar curvature of Σn is constant and greater than or equal to c. Then Σn is either totally umbilical, a (Riemannian) product of two totally umbilical constantly curved * Eudes L. de Lima [email protected] Henrique F. de Lima [email protected] 1
Unidade Acadêmica de Ciências Exatas e da Natureza, Universidade Federal de Campina Grande, Cajazeiras, Paraíba 58900‑000, Brazil
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Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba 58429‑970, Brazil
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submanifolds or possibly a flat manifold which is different from the above two types. The last case can happen only if c = 0. (If the ambient manifold is the Euclidean space, the last two cases cannot occur because of the compactness of Σn.) In the noncompact case, they extended the previous theorem when c = 0 characterizing such a hypersurface Σn as being a circular cylinder ℝp × 𝕊n−p . More precisely, they proved the following:
Theorem (Theorem 4 of [7]) Let Σn be a complete noncompact hypersurface in the Euclidean space with nonnegative curvature. Suppose that the scalar curvature of Σn is constant, then Σn is a generalized cylinder ℝp × 𝕊n−p . Their approach involves a careful study of an appropriate self-adjoint differential opera‑ tor introduced by themselves in [7], sometimes called of Cheng-Yau’s operator. Actually, this operator has become one of the most efficient to
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