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.
Categories: math.DG