I will start by reviewing older work, joint with Nicos Kapouleas, on the calculation of the Morse index of an infinite subfamily of Lawson's embedded minimal surfaces in the round 3-sphere. Then I will present recent bounds, obtained in collaboration with Alessandro Carlotto and Mario Schulz, concerning the Morse index of Hsiang's embedded rotationally invariant minimal hypersurfaces in the round 4-sphere.
Both projects make critical use of Courant's nodal domain theorem and related arguments applied by Montiel and Ros to the estimation of the index of complete minimal surfaces of finite total curvature in R3.
For the Lawson surfaces we employ such arguments in conjunction with a careful analysis of the Jacobi fields induced by ambient Killing fields. For the Hsiang hypersurfaces we instead exploit their relationship with a certain conformal Killing field. | 
