Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Abstract

Given a positive function F defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces M n immersed in the Euclidean space R n+1 , the so-called k-th anisotropic mean curvatures H F k , 0 ≤ k ≤ n. For fixed 0 ≤ r ≤ s ≤ n, a hypersurface M n of R n+1 is said to be (r, s, F )-linear Weingarten when its k-th anisotropic mean curvatures H F k , r ≤ k ≤ s, are linearly related. In this setting, we establish the concept of stability concerning closed (r, s, F )-linear Weingarten hypersurfaces immersed in R n+1 and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of F . For r = s and F ≡ 1, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [1].

On almost stable linear Weingarten hypersurfaces

“…Many of the classical characterizations of the geodesic hyperspheres have an analogue with the Wulff Shape as characteristic hypersurface, like anisotropic Hopf or Alexandrovtype theorems (see [14,15,16,18,19,20,23]). In [12], da Silva, de Lima and Velásquez gave an anisotropic analogue to their results in [11] and recover as particular cases the results of [6], [15] and [20]. This gives another characterization of the Wulff shape as the only hypersurface (up to translations and homotheties) which have linearly related anisotropic mean curvatures, without assuming that X(M ) is convex or embedded.…”

“…This gives another characterization of the Wulff shape as the only hypersurface (up to translations and homotheties) which have linearly related anisotropic mean curvatures, without assuming that X(M ) is convex or embedded. In the present note, we prove that the result of da Silva, de Lima and Velásquez in [12] can be improved with a weaker assumption, namely with almost (r, s, F )stability. Precisely, we prove the following: Theorem 1.1.…”

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

“…, n − 1} are just the hypersurfaces having constant (r + 1)-th mean curvature H r+1 . In recent years, several papers have been published showing the interest in understanding the geometry of the (r, s)-linear Weingarten hypersurfaces (see [2,3,14,15,23]). For instance, we can highlight that the author jointly with H. de Lima and A. de Sousa showed in [23, Section 3] that (r, s)-linear Weingarten closed hypersurfaces compact are critical points of the variational problem of minimizing a suitable linear combination…”

“…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]). Furthermore, when r = s ∈ {0, .…”

Some new characterizations of the Wulff shape

“…and some restricted a, b must be the Wulff shape. In 2018, da Silva et al [DDV18] showed that if there exist nonnegative real numbers…”