On the index of minimal hypersurfaces in S^n+1 with ₁<n

In this paper, we prove that a closed minimal hypersurface in with ₁<n has Morse index at least n+4, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in S³: a minimal torus in S³ has Morse index at least 5, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and r-minimal hypersurfaces in a sphere.