On the conformal Ein invariants

For a compact Riemannian $n$-manifold $(M,g)$ of positive scalar curvature, the capital $\Ein$ invariant of $g$ is defined to be the infinimum over $M$ of the quotient of the scalar curvature by the maximal eigenvalue of the Ricci curvature. This is a re-scale invariant and belongs to the interval $(0,n]$. For a positive conformal class $[g]$, we define the conformal invariant $\Ein([g]):=\sup\{\Ein(g): g\in [g]\}$. In this paper, we prove vanihing theorems for Betti numbers and for the higher homotopy groups of $M$ under optimal lower bounds on $\Ein([g])$ assuming that $g$ is locally conformally flat. We establish an inequality relating our invariant to Schoen-Yau conformal invariant $d(M,[g])$ from which we deduce a classification result for locally conformally flat manifolds with higher $\Ein([g])$. We show that the class of locally conformally flat manifolds with $\Ein([g])>k$ is stable under the operation of connected sums for $0<k<n-1.$\\ For a general positive conformal class, we prove in dimension $4$ an inequality relating $\Ein([g])$ to the first and second Yamabe invariants. Similar results are proved in this paper for an analogous conformal invariant, namely the small $\ein$ invariant.