Note on energy index and first eigenvalue of minimal surfaces in spheres
Matilde Gianocca
TL;DR
This work links the energy index of minimal immersions $\Sigma\subset S^n$ to the first eigenvalue $\lambda_1(\Sigma)$. Using Möbius fields $\xi_i$ as a canonical testing subspace, the authors compute the second variation of energy and establish a crucial identity $\sum_{i=1}^{n+1} D^2E(f\xi_i)= n\int_{\Sigma}|\nabla f|^2 - (2n-4)\int_{\Sigma} f^2$, connecting spectral data to energy variations. They prove that if $\lambda_1(\Sigma)<\frac{n-2}{2n}$, there exists a vector field $X$ orthogonal to the Möbius fields with $D^2E(X)<0$, giving a negative direction for the energy index; this extends to arbitrary codimension. The results provide a concrete spectral criterion that complements the known area-index relations and enhances understanding of index behavior for minimal surfaces in higher codimensions. This has potential implications for classifying minimal submanifolds by their energy index and for pursuing related conjectures on eigenvalues in minimal surface theory.
Abstract
A minimal immersion from a surface to $S^3$ can be viewed both as a critical point of the area and of the energy. Although no difference appears at first order, looking at the respective second variations unveils significant differences. It is well known that whenever the first eigenvalue satisfies $λ_1(Σ)\geq2$, the index is $\mathrm{ind}_E(Σ)\leq 4$. The converse implication is much more subtle. We prove that whenever $λ_1(Σ)<\frac{1}{6}$, there exists a vector field $X$, orthogonal to the four Möbius vector fields, with negative second variation. We also prove an arbitrary codimension version of this statement: any immersed minimal surface $Σ\subset S^n$ with first eigenvalue $λ_1(Σ)<\frac{n-2}{2n}$ admits a vector field $X$ orthogonal to the $n+1$ Möbius fields with negative second variation.