On the B\"ar-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem

Abstract

Let $M$ be a closed spin manifold of dimension $m\geq6$ equipped with a Riemannian metric $\ig$ and a spin structure $\sa$. Let $\lm_1^+(\tilde\ig)$ be the smallest positive eigenvalue of the Dirac operator $D_{\tilde\ig}$ on $M$ with respect to a metric $\tilde\ig$ conformal to $\ig$. The B\"ar-Hijazi-Lott invariant is defined by $\lm_{min}^+(M,\ig,\sa)=\inf_{\tilde\ig\in[\ig]}\lm_1^+(\tilde\ig)\Vol(M,\tilde\ig)^\frac{1}{m}$. In this paper, we show that \[ \lm_{min}^+(M,\ig,\sa)<\lm_{min}^+(S^m,\ig_{S^m},\sa_{S^m})=\frac m2\Vol(S^m,\ig_{S^m})^{\frac1m} \] provided that $\ig$ is not locally conformally flat. This estimate is a spinorial analogue to an estimate by T. Aubin, solving the Yamabe problem in this setting.