Overview
This paper addresses the stability index and Yau's conjecture concerning Carlotto-Schulz minimal hypertori. Specifically, it analyzes the multiplicity of an eigenvalue for the stability operator associated with the Carlotto and Schulz minimal embedding $X_{CS}^{n}:S^{n-1}\times S^{n-1}\times S^{1}\to S^{2n}$. The research includes a numerical verification of a proposed conjecture for the stability index of $X_{CS}^{n}$ and a numerical verification of Yau's conjecture on the first eigenvalue of the Laplacian for a defined range of $n$.
Research Context
For any closed minimal hypersurface $M$ embedded in the $N+1$-dimensional Euclidean sphere $S^{N+1}$, the value $-N$ is recognized as an eigenvalue of the stability operator. This fundamental property establishes a baseline for understanding the behavior of minimal hypersurfaces within these spherical geometries.
Approach
The research proceeded by calculating the multiplicity of the eigenvalue $-N$ for the Carlotto and Schulz minimal embedding $X_{CS}^{n}$. Subsequently, a conjecture proposing a specific formula for the stability index of $X_{CS}^{n}$ was formulated. This conjecture was then subjected to numerical verification for a range of $n$ values. Additionally, Yau's conjecture regarding the first eigenvalue of the Laplacian was numerically verified within a specified range of $n$.
Findings
- The multiplicity of the eigenvalue $-N$ for the stability operator for the Carlotto and Schulz minimal embedding $X_{CS}^{n}:S^{n-1}\times S^{n-1}\times S^{1}\to S^{2n}$ is determined to be at least $2n+1+n^2$.
- The researchers conjecture that if $n\ge 2$, the stability index of $X_{CS}^{n}$ is $\frac{1}{3} \left(n^3+9 n^2+11 n+3\right)$.
- For the specific case of the hypertorus in $S^4$ (where $n=2$), the conjectured stability index is $27$.
- Numerical verification of the conjectured stability index for $X_{CS}^{n}$ was performed for the first 100 values of $n$.
- Numerical verification indicated that Yau's conjecture on the first eigenvalue of the Laplacian holds when $2\le n\le 260$.