Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Jonatan Floriano da Silva, Henrique F. de Lima, Marco Antonio L. Velásquez
2018 Publicacions matemàtiques  
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
more » ... bility 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].
doi:10.5565/publmat6211805 fatcat:jnecu7a3z5dklo72vy5tne5nge