Let $M$ be a oriented surface with metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar function, corresponds to a conformal immersion $f:M\rightarrow \mathbb{R}^3$ (See Friedrich 03). It follows that the eigenvalue problem $D \phi =\lambda_i \phi$, where $\lambda_i\in \mathbb{R}$, gives some special immersion of this surface $f_i:M\rightarrow \mathbb{R}^3$. My questions is, if $f_1$, the immersion corresponding to the first eigenvalue, has some special meaning? Actually I'm highly interested in the Willmore energy of this immersion. I guess the this immersion would have a small Willmore energy from the inequality in (Bär 98): $$\lambda_1^2\leq \frac{\int_M H^2}{\mathrm{area}(M)}$$ However it might be too optimistic to expect that this immersion is a round sphere. Anyone has some insights for this immersion? Thank you very much.

Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar function, corresponds to a conformal immersion $f:M\rightarrow \mathbb{R}^3$ (See Friedrich 03). It follows that the eigenvalue problem $D \phi =\lambda_i \phi$, where $\lambda_i\in \mathbb{R}$, gives some special immersion of this surface $f_i:M\rightarrow \mathbb{R}^3$. My questions is, if $f_1$, the immersion corresponding to the first eigenvalue, has some special meaning? Actually I'm highly interested in the Willmore energy of this immersion. I guess the this immersion would have a small Willmore energy from the inequality in (Bär 98): $$\lambda_1^2\leq \frac{\int_M H^2}{\mathrm{area}(M)}$$ However it might be too optimistic to expect that this immersion is a round sphere. Anyone has some insights for this immersion? Thank you very much.