The stability of hypersurfaces revisited

Abstract

In this paper, we revisit the problem of characterize (r, s)-stable closed hypersurfaces immersed in a Riemannian space form, which was firstly established in Velásquez et al. (J Math Anal Appl 406:134-146, 2013). With a different approach of that used in the proof of the main theorem of Velásquez et al. (J Math Anal Appl 406:134-146, 2013), we complete its program showing that a closed hypersurface contained in the Euclidian space R n+1 and having higher order mean curvatures linearly related is (r, s)-stable if, and only if, it is a geodesic sphere of R n+1 .

Some new characterizations of the Wulff shape

“…, n}, and the relation H F r = aH F + b for some real constants a ≥ 0, b > 0 embedded in Euclidean space must be the Wulff shape, up to translations and homotheties. These results are anisotropic analogues of the results of [DDV16] and [D18] respectively, and both were obtained using a similar method to the isotropic case.…”

On the stability of the Wulff shape

“…They prove that the only compact (r, s)-stable hypersurfaces in the sphere or the hyperbolic space are geodesic hyperspheres. This result was recently extended for hypersurfaces of the Euclidean space by da Silva, de Lima and Velásquez [20].…”

“…The aim of this short note is to prove an anisotropic analogue to [20]. This extend to the anisiotropic (r, s)-stability the results of [18], [13] and [6] and give an other 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.…”

On almost stable linear Weingarten hypersurfaces

“…They prove that the only closed (r, s)-stable hypersurfaces in the sphere or the hyperbolic space are geodesic hyperspheres. This result was later extended for hypersurfaces of the Euclidean space by da Silva, de Lima and Velásquez [11]. On the other hand, during the last decade, an intensive interest has been brought to the study of hypersurfaces of Euclidean spaces in an anisotropic setting.…”