Let $M$ be a spherical <s>oriented</s> surface with Riemannian metric <s>and with trivial spin structure</s>. 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][1]). 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][2]):
$$\lambda_1^2\leq \frac{\int_M H^2}{\mathrm{area}(M)}$$
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.


  [1]: http://www.sciencedirect.com/science/article/pii/S0393044098000187
  [2]: https://link.springer.com/article/10.1023/A:1006550532236