Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

• Autores: Jonatan F. da Silva, Henrique F. de Lima, Marco A. L. Velásquez
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 62, Nº 1, 2018, págs. 95-111
• Idioma: inglés
• Enlaces
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• Resumen

  • Given a positive function F defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces Mn immersed in the Euclidean space Rn+1, the so-called k-th anisotropic mean curvatures HF k, 0 ≤ k ≤ n. For fixed 0 ≤ r ≤ s ≤ n, a hypersurface Mn of Rn+1 is said to be (r, s, F)-linear Weingarten when its k-th anisotropic mean curvatures HF 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 Rn+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.