Morse index of minimal products of minimal submanifolds in spheres

TL;DR

This paper investigates the Morse index and nullity of minimal product submanifolds in spheres, establishing conditions for their eigenfunction immersions and providing explicit index estimates for Clifford minimal submanifolds.

Contribution

It characterizes when minimal products are immersed by their first eigenfunctions and computes explicit Morse index and nullity formulas for Clifford minimal submanifolds.

Findings

Minimal product is immersed by first eigenfunctions iff initial submanifolds are.

Explicit Morse index formula for Clifford minimal submanifolds.

Nullity estimate for Clifford minimal submanifolds.

Abstract

Tang-Zhang, Choe-Hoppe, showed independently that one can produce minimal submanifolds in spheres via Clifford type minimal product of minimal submanifolds. In this note, 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 estimates of Morse index and nullity of the minimal product. In particular, we show that the Clifford minimal submanifold $\left(\sqrt{\frac{n_1}{n}}S^{n_1},\cdots,\sqrt{\frac{n_k}{n}}S^{n_k}\right)\subset S^{n+k-1}$ has index $(k-1)(n+k+1)$ and nullity $(k-1)\sum_{1\leq i<j\leq k}(n_i+1)(n_j+1)$ (with $n=\sum n_j$).