Stability index jump for constant mean curvature hypersurfaces of spheres

Abstract

It is known that the totally umbilical hypersurfaces in the n+1-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S n+1

Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^{3}$ and $\mathbb S^{3}$

“…Savo [37] used this fact to improve his estimates for the index of minimal hypersurfaces in the sphere. We would like to point out here that Perdomo and Brasil proved in [32] that if M is a compact CMC hypersurface in S n+1 which is not totally umbilical, then Ind w (M) ≥ n + 1, however the negative eigenvalues are not explicit and thus we cannot use it to improve our estimates.…”

“…On the other hand, there are few results in the literature about index estimates of other examples of CMC hypersufaces. To the best of the authors' knowledge, estimates were given by Lima-Sousa Neto-Rossman [26] and Rossman [35] for CMC tori in R 3 , by Perdomo-Brasil [32] for CMC hypersurfaces in the sphere, but in terms of the dimension, and by Rossman-Sultana [36] and Cañete [13] for CMC tori in S 3 .…”

“…In the case H = 0, Perdomo and Brasil [32] proved that if M ⊂ S n+1 is a compact CMC hypersurface not totally umbilical, then Ind w (M) ≥ n + 1 (see also [2] for a nice survey). Recently, Alias and Piccione [3] showed the existence of infinite sequences of isometric embeddings of tori with constant mean curvature in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.…”

A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres

“…These two family of functions {l v } v∈R n+2 and {f v } v∈R n+2 are very important because they can be defined for any oriented immersion on the sphere and they have been widely used to construct test functions to study the spectrum of operators on M such as the Laplace operator and the stability operator when M is minimal, M has constant mean curvature or M has constant scalar curvature. See for example [11], [12], [13], [14], [9], [6] and [4].…”

The stability index of hypersurfaces with constant scalar curvature in spheres

“…gave estimates about the weak stability index of a compact hypersurface with constant mean curvature in S n+1 (1) under the assumption that the scalar curvature is constant in [2,3]. In [11], Perdomo and Brasil proved that the weak stability index of a compact hypersurface with constant mean curvature in S n+1 (1) is greater than or equal to n + 1 if the hypersurface is non-totally umbilical.…”