2020/09/24 by Mohammed Larbi Labbi, Labbi, Mohammed Larbi
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2009.11601
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∈ [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