# The first eigenfunction of Dirac operator for surface

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.

If $M$ is $S^2$, then every solution of $D\phi= \rho \phi$ as above comes from a conformal immersion, potentially with branching points of even order (the "even order" condition is severly neglected in the literature!). This is no longer in the case of higher genus surfaces in contrast to what you indicate above.
• The historical remarks are interesting! Sorry, for this moment I'm only interested in the $S^2$ case and my previous assumptions were too general. Now I've corrected my questions. I happen to know a bit German. Your PHD thesis is a very nice source, I will read it slowly :) – Z. Ye Jun 5 '17 at 9:14