Almost harmonic spinors (Q990214)

The authors prove the following theorem. For any \((n\geq 2)\) \(n\)-dimensional closed spin manifold \(M^n\) not diffeomorphic to \(\mathbb{S}^2\) there exists a sequence \((g_p)_{p\in\mathbb{N}}\) of Riemannian metrics on \(M^n\) such that \[ \lambda^+_1(D^2_{g_p})\text{Vol}(M^n, g_p)^{{2\over n}}@>(p\to\infty)>> 0. \] Here \(D_g\) denotes the spin Dirac operator associated to a Riemannian metric \(g\) and \(\lambda_1(D^2_g)\) and \(\lambda^+_1(D^2_g)\) are the smallest and the smallest positive eigenvalue of \(D^2_g\), respectively. As an application, the authors compare the Dirac spectrum with the conformal volume.