On two constants of a positive conformal class
Tom 176 / 2024
Streszczenie
For a compact Riemannian $n$-manifold $(M,g)$ of positive scalar curvature, the metric Ein constant 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 operator. This is a re-scale invariant and it lies in the interval $(0,n]$. For a positive conformal class $[g]$, we define the conformal constant $\mathrm{Ein}([g]):=\sup \{\mathrm{Ein}(g): g\in [g]\}$. In this paper, we prove vanishing theorems for Betti numbers and for higher homotopy groups of $M$ under optimal lower bounds on $\mathrm{Ein}([g])$ assuming that $g$ is locally conformally flat. We establish an inequality relating the above conformal invariant to the Schoen–Yau conformal invariant $d(M,[g])$, from which we deduce a classification result for locally conformally flat manifolds with sufficiently high $\mathrm{Ein}([g])$. We show that the class of locally conformally flat manifolds with $\mathrm{Ein}([g]) \gt k$ is stable under the operation of connected sum provided that $0 \lt k \lt n-1.$
For a general positive conformal class, we prove in dimension $4$ an inequality relating $\mathrm{Ein}([g])$ to the first and second Yamabe invariants. Similar results are proved for an analogous conformal constant, the small ein invariant.
