The region of the unit Euclidean sphere that admits a class of (r,s)-linear Weingarten hypersurfaces
DOI:
https://doi.org/10.17268/sel.mat.2023.02.05Keywords:
unit Euclidean space, (r, s)-linear Weingarten hypersurfaces, upper (lower) domain enclosed by the geodesic sphere of unit Euclidean space of level τ0, strong stability, geodesic spheresAbstract
In the unit Euclidean sphere Sn+1, we deal with a class of hypersurfaces that were characterized in [23] as the critical points of a variational problem, the so-called (r, s)-linear Weingarten hypersurfaces (0 ≤ r ≤s ≤ n−1); namely, the hypersurfaces of Sn+1 that has a linear combination arHr+1+・ ・ ・+asHs+1 of their higher order mean curvatures Hr+1 and Hs+1 being a real constant, where ar, . . . , ar are nonnegative real numbers (with at least one non zero). By assuming a geometric constraint involving the higher order mean curvatures of these hypersurfaces, we prove a uniqueness result for strongly stable (r, s)-linear Weingarten hypersurfaces immersed in a certain region determined by a geodesic sphere of Sn+1. We also establish a nonexistence result in another region of Sn+1 for strongly stable Weingarten (r, s)-linear hypersurfaces.
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