A spin-conformal lower bound of the first positive Dirac eigenvalue
Let \(\sigma\) be a spin structure on a compact Riemannian manifold \((M,g_0)\) of dimension \(m\). Let \(\lambda_k(g,\sigma)\) be the \(k\)-th eigenvalue of the spin Laplacian (i.e. the square of the associated Dirac operator \(A(g,\sigma)\)) for any metric \(g\in[g_0]\) which is conformally equivalent to \(g_0\). Set \[ Q_k(M,[g_0],\sigma):=\inf_{g\in[g_0]}\{\lambda_k(h,\sigma)\text{vol}(M,h)^{2/m}\}; \] one introduces an appropriate factor involving the volume to correct for eigenvalue changes under trivial rescalings. Since the Dirac operator transforms conformally, \(\dim\ker A\) is a conformal invariant. Thus \(\lambda_{\dim\ker A+1}(g,\sigma)\)is the first non-zero eigenvalue of the spin Laplacian. If the Yamabe number \(\mu(M,[g_0])\) is positive, \textit{O. Hijazi} [Commun. Math. Phys. 104, 151-162 (1986; Zbl 0593.58040)] showed \[ Q_1(M,[g_0],\sigma)\geq{n\over 4(n-1)}\mu(M,[g_0])>0 . \] It was shown by \textit{J. Lott} [Pac. J. Math. 125, 117-126 (1986; Zbl 0605.58044)] that if \(A\) is invertible, then \(Q_1(M,[g_0],\sigma)>0\) even if the Yamabe number is zero or negative. In the present article the author extends this result to show that \[ Q_{\dim\ker A+1}(M,[g_0],\sigma)>0. \] If, for example, one restricts to metrics with \(vol(M,h)=1\), then this shows there exists a universal lower bound for the first non-zero eigenvalue on a given conformal class. It is an open problem to find explicit lower bounds in the case \(\mu\leq 0\) and \(\dim\ker A>0\) in terms of geometric data. The proof uses a conformal blow-up of a sphere in a small neighborhood of a given point.
scientific article; zbMATH DE number 1970786