Strongly Stable Linear Weingarten Hypersurfaces Immersed in the Hyperbolic Space

Abstract

In this article, we establish the notion of strong stability related to closed linear Weingarten hypersurfaces immersed in the hyperbolic space. In this setting, initially we show that geodesic spheres are strongly stable. Afterwards, under a suitable restriction on the mean and scalar curvatures, we prove that if a closed linear Weingarten hypersurface into the hyperbolic space is strongly stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in a chronological future (or past) of the de Sitter space.Mathematics Subject Classification. Primary 53C42; Secondary 53B30, 53C50, 53Z05, 83C99.

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

“…On the other hand, a natural extension of the hypersurfaces Σ n ↬ S n+1 with constant mean curvature H or constant second mean curvature H 2 are those ones whose curvatures H and H 2 obey a linear relation of the type a 0 H + a 1 H 2 = constant, for some real constants a 0 and a 1 . These hypersurfaces are called in the literature as linear Weingarten hypersurfaces (see, for instance, [5,6,7,12,16,17,18]). A class that extends such hypersurfaces is given by the so-called generalized linear Weingarten hypersurfaces, namely, those hypersurfaces whose higher order mean curvatures H r+1 and H s+1 (for entire numbers r and s such that 0 ≤ r ≤ s ≤ n − 1) satisfy the linear condition a r H r+1 + • • • + a s H s+1 = constant, for some real numbers a r , .…”

“…Remark 2.1. Taking into account the relation between H 2 and the normalized scalar curvature R given in (2.4), we observe from (2.10) that the (0, 1)-linear Weingarten hypersurfaces x : Σ n ↬ S n+1 are called simply linear Weingarten hypersurfaces, and there is a vast recent literature treating the problem of characterizing these hypersurfaces (see, for instance, [5,6,7,12,16,17,18]). It is because of this observation that the hypersurfaces described in Definition 2.1 are also called, in the current literature, the generalized linear Weingarten hypersurfaces (see [2,3,14,15,23]).…”

On the index of stability of (r, s)-linear Weingarten Clifford tori

“…for some real constants a 0 and a 1 (at least one of them nonzero). The Clifford toi T n1,n2 ρ1, ρ2 ↬ S n+1 that satisfy (2.11) belongs to a class of hypersurfaces called the linear Weingarten, and there is a vast recent literature treating the problem of characterizing these hypersurfaces (see, for instance, [3,4,5,11,16,17,18]). This class of Clifford tori contains those that are minimal (when a 1 = 0 in (2.11)), as well as those that are 1-minimal (when a 0 = 0 in (2.11)).…”

“…On the other hand, a natural extension of the hypersurfaces Σ n ↬ S n+1 with constant mean curvature H or constant second mean curvature H 2 are those ones whose curvatures H and H 2 obey a linear relation of the type a 0 H + a 1 H 2 = constant, for some real constants a 0 and a 1 . These hypersurfaces are called in the literature as linear Weingarten hypersurfaces (see, for instance, [3,4,5,11,16,17,18]). A class that extends such hypersurfaces is given by the so-called generalized linear Weingarten hypersurfaces, namely, those hypersurfaces whose higher order mean curvatures H r+1 and H s+1 (for entire numbers r and s such that 0 ≤ r ≤ s ≤ n − 1) satisfy the linear condition a r H r+1 + • • • + a s H s+1 = constant, for some real numbers a r , .…”