Index estimates for closed minimal submanifolds of the sphere

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$ -forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.

Transient dynamics by continuous-spectrum perturbations in stratified shear flows

The transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.