The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions

Let us fix a conformal class $[g_0]$ and a spin structure $\sigma$ on a compact manifold $M$. For any $g\in [g_0]$, let $\lambda^+_1(g)$ be the smallest positive eigenvalue of the Dirac operator $D$ on $(M,g,\sigma)$. In a previous paper we have shown that $$\lambda_{min}(M,g_0,\sigma):=\inf_{g\in [g_0]} \lambda_1^+(g)\vol(M,g)^{1/n}>0.$$ In the present article, we enlarge the conformal class by certain singular metrics. We will show that if $\lambda_{min}(M,g_0,\sigma)<\lambda_{min}(S^n)$, then the infimum is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial differential equations involving a nonlinearity with critical exponent: $$D\phi= \lambda |\phi|^{2/(n-1)}\phi.$$ The solution of this problem has many analogies to the solution of the Yamabe problem. However, our reasoning is more involved than in the Yamabe problem as the eigenvalues of the Dirac operator tend to $+\infty$ and $-\infty$. Using the Weierstra\ss{} representation, the solution of this equation in dimension 2 provides a tool for constructing new periodic constant mean curvature surfaces.