The Morse index of minimal products of minimal submanifolds in spheres

Tang and Zhang (2020) and Choe and Hoppe (2018) showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds. In this paper, we show that the minimal product is immersed by its first eigenfunctions (of its Laplacian) if and only if the two beginning minimal submanifolds are immersed by their first eigenfunctions. Moreover, we give the estimates of the Morse index and the nullity of the minimal product. In particular, we show that the Clifford minimal submanifold has the index ( k − 1)( n + k + 1) and the nullity ( k − 1) ∑ 1⩽ i < j ⩽ k ( n i + 1)( n j + 1) (with n = ∑ n j ).