Higher order mean curvature estimates for bounded complete hypersurfaces

Abstract

We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.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.Let f : M n →M n+1 be a codimension one isometric immersion between complete Riemannian manifolds. Assume that the hypersurface lies inside a closed geodesic ball BM (r) of radius r and center o ∈M n+1 and that 0 < r < min{injM (o), π/2 √ b} where injM (o) is the injectivity radius at o and π/2 √ b is replaced by +∞ if b ≤ 0. Suppose also that there is a point p 0 ∈ M n such that f (p 0 ) ∈ SM (r) where SM (r) is the boundary of BM (r). In the context of this paper, this is a slightly weaker assumption than asking M n to be compact. Let K rad M denote the radial sectional curvatures in BM (r) along geodesics issuing from the center and assume that K

Analytic and geometric properties of generic Ricci solitons

“…and weakly on all of M . Proceeding in a way similar to that in the proof of Theorem 3 in [2] we have the following proposition that improves on [13].…”

A general form of the weak maximum principle and some applications

“…However, to guarantee its validity ones needs to assume the existence of an elliptic point (see [4] for details).…”

Higher order mean curvature estimates for complete hypersurfaces into horoballs

“…We should remark that Bessa et al also obtained higher order mean curvature estimates via eigenvalue estimates of the L r operator, see [6]. In [3], L. Alias, M. Dajczer and M. Rigoli extended mean curvature estimates of Jorge and Xavier [21] and Ranjbar-Motlagh to higher order mean curvature estimates in the following result.…”

“…Theorem 2.1 (Alias-Dajczer-Rigoli [3]). Let ϕ : M → N be an isometrically immersed, complete hypersurface of a Riemannian (n + 1)-manifold N .…”

“…In this paper we need a slightly more general version of the weak maximum principle for the operators L r , introduced by Alias, Dajczer and Rigoli [3]. Let M be a (complete) Riemannian manifold and let L r (u) = Tr(P r hess u) be a semi-elliptic linear operator, u ∈ C 2 (M ).…”