K Ahler Parabolicty and the Euler Number of Compact Manifolds of Non-positive Sectional Curvature

Let M 2n be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if M 2n is homeomorphic to a KK ahler manifold, then its Euler number satisses the inequality (?1) n (M 2n) 0. Introduction The results of this paper are related to a well-known problem, attributed sometimes to Hopf and sometimes to Chern, to the eeect that the Euler number (M 2n) of a compact Riemannian manifold M 2n of negative sectional curvature must satisfy the inequality (?1) n (M 2n) > 0. This conjecture is true in dimensions 2 and 4 Ch] and it has been veriied in the KK ahler case for all n by Gromov G] and Stern S] (the work in S] also uses results of Greene and Wu; see GW], p.183-215). Gromov's arguments are rather general and establish the following result: \Let M 2n be a compact Riemannian manifold of negative curvature. If M 2n is homotopy equivalent to a compact KK ahler manifold then (?1) n (M 2n) > 0"; see G], Theorem 0.4.A and Example (a), p.265. A companion conjecture asserts that if the sectional curvature of a Riemannian manifold M 2n is assumed to be only non-positive, then the Euler number must satisfy (?1) n (M 2n) 0. Again, this second conjecture is known to be true in dimensions two and four Ch]. The aim of this paper is to establish its validity for all n in the KK ahler case, thus complementing the above result of Gromov: