Note on energy index and first eigenvalue of minimal surfaces in spheres

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 $\lambda_1(\Sigma)\geq2$, the index is $\mathrm{ind}_E(\Sigma)\leq 4$. The converse implication is much more subtle. We prove that whenever $\lambda_1(\Sigma)<\frac{1}{6}$, there exists a vector field $X$, orthogonal to the four M\"obius vector fields, with negative second variation. We also prove an arbitrary codimension version of this statement: any immersed minimal surface $\Sigma\subset S^n$ with first eigenvalue $\lambda_1(\Sigma)<\frac{n-2}{2n}$ admits a vector field $X$ orthogonal to the $n+1$ M\"obius fields with negative second variation.