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

Abstract

Recently, for any n>1, Carlotto and Schulz showed the existence of a minimal embedding in the 2n-dimensional unit sphere. In this paper, we show that the stability index of these embedded minimal hypersurfaces is at least n^2+4n+3. We also show that Yau's conjecture holds for these examples if and only if the solution of the differential equation z''(t)+a_n(t)z'(t)+(2n-1)z(t)=0 with z(0)=1 and z'(0)=0 satisfies z'(T)>0. Here,T and the T-periodic function a_n(t) are determined in terms of the functions defining the minimal immersion.

This version corrects minor errors in the first submission. In particular, the multiplicity formula in Theorem 3.3 has been fixed (product instead of sum), and the lower bound for the stability index is updated to n^2+4n+3 (instead of 3n+6). The abstract and introduction have been made consistent with this correction. No other results are affected