On two constants of a positive conformal class
Tom 176 / 2024
Streszczenie
For a compact Riemannian -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.