Abstract
To a given immersion \({i:M^n\to \mathbb S^{n+1}}\) with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|2. We prove the existence of a constant C n (R) depending on R and n so that R ≥ 1 and sup |A|2 = C n (R) imply that the hypersurface is a H(r)-torus \({\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}\). For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C n (R) there is a complete hypersurface in \({\mathbb S^{n+1}}\) with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alencar H., do Carmo M.P.: Hypersurfaces with constant mean curvature in spheres. Proc. AMS 120(4), 1223–1229 (1994)
Cheng Q.M.: Hypersurfaces in a unit sphere \({\mathbb S^{n+1}}\) with constant scalar curvature. J. Lond. Math. Soc. 64(2), 755–768 (2001)
Cheng Q.M., Shu S., Suh Y.J.: Compact hypersurfaces in a unit sphere. Proc. Royal Soc. Edinburgh 135, 1129–1137 (2005)
Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
do Carmo M.P., Dajczer M.: Rotational hypersurfaces in spaces of constant curvature. Trans. Am. Math. Soc. 277(2), 685–709 (1983)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1977, 1983)
Leite M.L.: Rotational hypersurfaces of space forms with constant scalar curvature. Manuscr. Math. 67, 285–304 (1990)
Li H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305, 665–672 (1996)
Li H.: Global rigidity theorems of hypersurfaces. Ark. Mat. 35, 327–351 (1997)
Okumura M.: Hypersurfaces and a pinching problem on the second fundamental tensor. Am. J. Math. 96, 207–213 (1974)
Omori H.: Isometric Immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
Otsuki T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature. Am. J. Math. 92, 145–173 (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Alekseevsky.
A. Brasil Jr. was partially supported by CNPq, Brazil. A. G. Colares was partially supported by FUNCAP, Brazil. O. Palmas was partially supported by CNPq, Brazil and DGAPA-UNAM, México, under Project IN118508.
Rights and permissions
About this article
Cite this article
Brasil, A., Colares, A.G. & Palmas, O. Complete hypersurfaces with constant scalar curvature in spheres. Monatsh Math 161, 369–380 (2010). https://doi.org/10.1007/s00605-009-0128-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-009-0128-9