Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

Abstract

We find many examples of compact Riemannian manifolds (M, g) whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric g is replaced with g ′ in a neighbourhood of g. Our examples (M, g) consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups SU(n) for n ≤ 17, and Sp(n) for all n; and all quaternionic Grassmannians.2010 Mathematics Subject Classification. 49Q05, 53A10, 53C35, 53C40.

Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

“…In both works, the authors make use of the parallel canonical orthonormal frame in an appropriate R d . Related results were obtained by Mendes Radeschi [19], Gorodski, Mendes and Radeschi [14], and Chao Li [17]. These results strongly support Schoen and Marques-Neves conjecture [18,20], which claims that if the ambient space M n+1 is complete and has positive Ricci curvature, then there exists a positive constant C, depending only on M , such that for any closed minimal hypersurface Σ in M it holds that…”

Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces

“…When it comes to lower index bounds, starting with an idea of A. Ros [46] there have been many articles verifying the conjecture of Marques-Neves-Schoen (1) mentioned earlier for ambient spaces (M, g) carrying special metrics: see [46,3] for flat tori, A. Savo [47] for round spheres, F. Urbano [53] for S 1 ×S 2 , L. Ambrozio-Carlotto-Sharp [2] for compact rank one symmetric spaces and more, C. Gorodski-R. A. E. Mendes-M.Radeschi [21] etc. These papers are based on subtle refinements of Ros' method and consequently their results do not depend on an area upper bound but require the metric to be very symmetric.…”

Index estimates for closed minimal submanifolds of the sphere

“…Finally, we highlight that Savo in [18] performed an ingenious trick to obtain a comparison theorem between the spectrums of the stability operator acting on functions and the Hodge–Laplacian acting on -form of closed orientable minimal hypersurfaces of , and such comparison implies a lower estimate of the Morse index by a linear function of its first Betti's number. This technique was refined in many directions, see for instances the breakthrough done in [2, 11]. Also, due to the fruitfulness of this technique, very recently such method was successfully adapted for the constant mean curvature and weighted minimal hypersurfaces settings, see for instance [1, 3, 4, 9, 10].…”