Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres

TL;DR

The paper develops a framework to study the spectrum of the Laplacian and the Jacobi (stability) operator on generalized rotational minimal hypersurfaces of spheres, by expressing the mean curvature of the ambient rotated hypersurface $M$ in terms of the principal curvatures of the generating piece $M_0$ and by reducing spectral problems to one-dimensional ODEs via spherical and Fourier decompositions. It treats two main cases: $M_0$ as a surface of revolution in $\mathbb{R}^3$ and as a higher-codimension hypersurface of revolution, deriving explicit forms for the Laplace and stability operators and establishing a union-of-ODE-spectra description. The work provides a practical numerical pipeline to compute eigenvalues, and demonstrates it on a minimal embedded example in $S^6$, yielding detailed spectra, a stability index of $77$, and a nullity of $14$, along with additional embedded examples in various dimensions. These results illustrate the feasibility of numerically analyzing spectral data for complex minimal hypersurfaces in spheres and contribute to understanding Yau-type conjectures and stability phenomena in this geometric setting.

Abstract

Let $M\subset S^{n+1}$ be the hypersurface generated by rotating a hypersurface $M_0$ contained in the interior of the unit ball of $\mathbb{R}^{n-k+1}$. More precisely, $M=\{(\sqrt{1-|m|^2}\, y, m):y\in S^k, m\in M_0\}$. We derive the equation for the mean curvature of $M$ in terms of the principal curvatures of $M_0$. For the particular case when $M_0$ is a surface of revolution in $\mathbb{R}^3$, we provide a method for finding the eigenvalues of the Laplace and stability operators. To illustrate this method, we consider an example of a minimal embedded hypersurface in $S^6$ and numerically compute all the eigenvalues of the Laplace operator less than 12, as well as all non-positive eigenvalues of the stability operators. For this example, we show that the stability index (the number of negative eigenvalues of the stability operator, counted with multiplicity) is 77, and the nullity (the multiplicity of the eigenvalue $λ=0$ of the stability operator) is 14. Similar results are found in the case where $M_0$ is a hypersurface in $\mathbb{R}^{l+2}$ of the form $(f_2(u)z, f_1(u))$, with $z$ in the $l$-dimensional unit sphere $S^l$. Carlotto and Schulz have found examples of embedded minimal hypersurfaces in the case where $M_0=S^k\times S^1$.

Figures (12)

• Figure 1: Solution of the system with $k=3$ and $a_0=0.14971331$.
• Figure 2: Graph of the surface $M_0$. The torus on the right is the rotation around the $x_1$-axis of the closed curve on the left.
• Figure 3: Graph of the function $\delta_{00}$ associated with the operator $L_{00}$
• Figure 4: Graph of the first three eigenfunctions of $L_{00}$
• Figure 5: Graph of the functions $\delta_{10}$ associated with the operator $L_{10}$, $\delta_{01}$ associated with the operator $L_{01}$ and $\delta_{11}$ associated with the operator $L_{11}$.

Theorems & Definitions (27)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3