Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

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

Sharp upper estimates for the first eigenvalue of a Jacobi type operator

Our purpose in this article is to obtain sharp upper estimates for the first positive eigenvalue of a Jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed either in the Euclidean space or in the Euclidean sphere.Brazilian National Research Council (CNPq): grant 303977/2015-9 y grant 308757/2015-7.peerReviewe

Characterizations of complete linear Weingarten spacelike submanifolds in a locally symmetric semi-Riemannian manifold

In this paper, we deal with n-dimensional complete spacelike submanifolds Mn with flat normal bundle and parallel normalized mean curvature vector immersed in an (n + p)-dimensional locally symmetric semi-Riemannian manifold L_p^(n+p) of index p obeying some standard curvature conditions which are naturally satisfied when the ambient space is a semi-Riemannian space form. In this setting, we establish sufficient conditions to guarantee that, in fact, p = 1 and Mⁿ is isometric to an isoparametric hypersurface of L_1^(n+1)having two distinct principal curvatures, one of which is simple.The first author is partially supported by CAPES, Brazil. The second author is partially supported by CNPq, Brazil, grant 303977/2015- 9. The fourth author is partially supported by CNPq, Brazil, grant 308757/2015-7.peerReviewe

Sharp upper estimates for the first eigenvalue of a Jacobi type operator

Our purpose in this article is to obtain sharp upper estimates for the first positive eigenvalue of a Jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed either in the Euclidean space or in the Euclidean sphere.Brazilian National Research Council (CNPq): grant 303977/2015-9 y grant 308757/2015-7.peerReviewe

Gap type results for spacelike submanifolds with parallel mean curvature vector

We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650)