Hypersurfaces with constant mean curvature in a real space form

Authors

SHICHANG SHU
SANYANG LIU

DOI

10.3906/mat-0904-59

Abstract

Let M^n be an n\(n \geq 3)-dimensional complete connected and oriented hypersurface in M^{n+1}(c)(c \geq 0) with constant mean curvature H and with two distinct principal curvatures, one of which is simple. We show that (1) if c=1 and the squared norm of the second fundamental form of M^n satisfies a rigidity condition (1.3), then M^n is isometric to the Riemannian product S^1(\sqrt{1-a^2}) \times S^{n-1}(a); (2) if c=0, H \neq 0 and the squared norm of the second fundamental form of M^n satisfies S \geq n^2H^2/(n-1), then M^n is isometric to the Riemannian product S^{n-1}(a)\times R or S^1(a) \times R^{n-1}

Keywords

Hypersurface, scalar curvature, mean curvature, principal curvature

First Page

301

Last Page

310

Recommended Citation

SHU, SHICHANG and LIU, SANYANG (2011) "Hypersurfaces with constant mean curvature in a real space form," Turkish Journal of Mathematics: Vol. 35: No. 2, Article 11. https://doi.org/10.3906/mat-0904-59
Available at: https://journals.tubitak.gov.tr/math/vol35/iss2/11