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.