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

TL;DR

This paper investigates the stability index of minimal hypersurfaces in spheres constructed by Carlotto and Schulz, and explores the validity of Yau's conjecture for these examples through differential equation analysis.

Contribution

It establishes a lower bound for the stability index and characterizes when Yau's conjecture holds for these minimal hypertori based on a specific differential equation.

Findings

Stability index is at least n^2+4n+3.

Yau's conjecture holds if and only if a certain differential equation solution satisfies z'(T)>0.

The differential equation involves a T-periodic coefficient linked to the minimal immersion.

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.