On the conformal Ein invariants

TL;DR

This paper introduces and analyzes the conformal Ein invariant for compact Riemannian manifolds, establishing vanishing theorems, inequalities, and classification results, especially for locally conformally flat manifolds with positive scalar curvature.

Contribution

It defines the conformal Ein invariant, proves vanishing theorems and stability results, and relates it to other invariants like the Schoen-Yau and Yamabe invariants, providing new insights into conformal geometry.

Findings

Vanishing theorems for Betti numbers and homotopy groups under lower bounds on the invariant.

Classification of locally conformally flat manifolds with high Ein invariant.

Stability of certain manifold classes under connected sums.

Abstract

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.