The stability index and Yau's conjecture for Carlotto-Schulz minimal hypertori, part II

Abstract

For any closed minimal hypersurface $M$ in the $N+1$-dimensional Euclidean sphere $S^{N+1}$, $-N$ is an eigenvalue of the stability operator. In this paper we show that the multiplicity of this eigenvalue for the Carlotto and Schulz minimal embedding $X_{CS}^{n}:S^{n-1}\times S^{n-1}\times S^{1}\to S^{2n}$ is at least $2n+1+n^2$. We conjecture that if $n>2$, then the stability index of $X_{CS}^{n}$ is $\frac{1}{3} \left(n^3+9 n^2+11 n+3\right)$ and for the hypertorus in $S^4$ (case $n=2$) the stability index is $27$. We numerically verify the conjecture for the first 100 values of $n$. We also numerically verify that Yau's conjecture on the first eigenvalue of the Laplacian holds when $2\le n\le 260$.