Metrics with harmonic spinors (Q2565227)

The main result is stated by the author as follows: Theorem A. Let \(M^n\) be a closed spin manifold of dimension \(n\), \(n\equiv 3\text{ mod }4\). Let a spin structure on \(M\) be fixed. Then there exists a Riemannian metric on \(M\) such that the corresponding Dirac operator has a nontrivial kernel, i.e., there are nontrivial harmonic spinors. The proof is complex and original. It involves the computation of the Dirac spectrum of the Berger spheres \(S^{2n+1}\) as well as: Theorem B. Let \(M_1\) and \(M_2\) be two closed Riemannian spin manifolds of odd dimension \(n\). Let \(D_1\) and \(D_2\) be the two Dirac operators. Let \(\varepsilon>0\) and let \(\Lambda>0\) be such that \(-\Lambda\), \(\Lambda\not\in\text{spec}(D_1)\cup\text{spec}(D_2)\). Write i) \((\text{spec}(D_1)\cup \text{spec}(D_2))\cap(-\Lambda, \Lambda)=\{\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_k\}\). Then there exists a Riemannian metric on the connected sum \(M_1\# M_2\) with Dirac operator \(D\) such that the following holds: ii) \(\text{spec}(D)\cap(-\Lambda,\Lambda)= \{\mu_1\leq\mu_2\leq\cdots\leq \mu_k\}\); iii) \(|\lambda_i-\mu_i|< \varepsilon\). To prove Theorem B, the author studies the distribution of the \(L^2\)-norm of the eigenspinors over cylindrical manifolds, in particular over Euclidean annuli.