Rigidity of spheres in Riemannian manifolds and a non-embedding theorem

Higher order mean curvature estimates for bounded complete hypersurfaces

“…Estimates for the k-mean curvatures H k of higher order of a compact hypersurface in a complete Riemannian manifold have been subsequently obtained by Vlachos [14], Veeravalli [13], Fontenele-Silva [9], Roth [12] and Ranjbar-Motlagh [11]. In this paper, we generalize a result given in the latter that we describe next.…”

“…First observe that u * cannot be attained at a point x 0 ∈ M n , for otherwise x 0 ∈ B η but, since P is positive semi-definite, then q(x 0 )Lu(x 0 ) ≤ 0 thus contradicting (11). Set…”

Higher order mean curvature estimates for complete hypersurfaces into horoballs

“…Jorge and Xavier [21] proved mean curvature estimates for compact submanifolds of Hadamard manifolds given in terms of the extrinsinc radius. Ranjbar-Motlagh [31] proved higher order mean curvature estimates for bounded hypersurfaces assuming certain extra conditions on the immersion. We should remark that Bessa et al also obtained higher order mean curvature estimates via eigenvalue estimates of the L r operator, see [6].…”

“…More generally, give higher order mean curvature estimates for complete submanifolds subject to extrinsic bounds. For compact hypersurfaces ϕ : M → N , higher order mean curvature estimates were obtained in terms of the extrinsic radius of ϕ(M ) and the geometry of N by various authors, see [6], [13], [31], [33], [36] and [37]. The next stage is to consider complete non-compact hypersurfaces under various curvature constraints and prove estimates for higher order mean curvatures.…”