The first eigenfunction of Dirac operator for surface

Question

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.

Answer 1

First some historical comments: The first sufficiently clear reference that I know for the spinorial Weierstrass is a preprint by Rob Kusner and Nick Schmitt link. From the point of view of the results, Friedrich does not add much, the value of Friedrich's article mostly lies in its very explicit presentation in invariant language. Friedrich's article was also available to experts after Baer's article. There is some discussion about the question whether the spinorial Weierstrass representation was known before Kusner-Schmitt. Obviously, by performing some extra computations one can read off the spinorial Weierestrass representationalready in older work, and depending on how much extra work should be done, one gets another inventor. Taimanov points to Eisenhart, but I do not find it visible from Eisenhart.

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.

There is also another unclear point in your question: it is not clear what is meant by a "trivial spin structure". On the sphere there is a unique one. If M is of genus at least 1, then it carries several spin structures that are the boundary of a 3-dimensional compact spin manifold, let us call them "bounding spin structures", and there is at least one spin structure which is not such a boundary. On a torus there is one non-bounding spin structure, which is often called the trivial one (including my early articles), but it is the only spin structure on the torus which defines a non-trivial element in the spin bordism group, so I try to avoid the word "trivial" in this context. On surfaces of genus at least 2 there are several non-bounding and several bounding ones, and there is no particular one which would deserve the name trivial.

So you should formulate your question more precisely in order that it can be answered more precisely.

Many questions related were considered within my PhD thesis, German, available on link and in my article with C. Bär Dirac eigenvalue estimates on surfaces Math. Z. 240, 423-449 (2002).