Let M 2n be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if M 2n is homeomorphic to a Kähler manifold, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. M 2n of negative sectional curvature must satisfy the inequality (−1) n χ(M 2n ) > 0. This conjecture is true in dimensions 2 and 4 [Ch] and it has been verified in the Kähler 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 Kähler manifold then (−1) n χ(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 χ(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 Kähler case, thus complementing the above result of Gromov:Main Theorem. Let M 2n be a compact Riemannian manifold of non-positive curvature. If M 2n is homeomorphic to a Kähler manifold, then the Euler number of M 2n satisfies the inequality (−1) n χ(M 2n ) ≥ 0.Remark. The proof of the Main Theorem works if the two manifolds are assumed to be only homotopy equivalent but, in view of Farrell and Jones [FJ], the man-

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger constant h(X) may be zero if any of the above assumption on X is removed.Applications of this isoperimetric estimate include the solvability of the Dirichlet problem at infinity for non-compact Gromov-hyperbolic manifolds X above. In addition, we show that the Martin boundary ∂ ∆ X of such a space X is homeomorphic to the geometric boundary X(∞) of X at infinity.

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We would like to thank Professors T. Ohsawa and K. Diederich for pointing out an error in our paper [CSW]. The inequality (4.18) in the proof of Proposition 4.5 is not valid. Step 2 in the proof of Proposition 4.5 should be changed to the following stronger statement using the unweighted∂-Neumann operator. Proposition 4.5 still holds after the correction with other steps unchanged.

We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi (J. Differ. Geom. 56:1-66, 2000; Math. Ann. 333: 131-155, 2005) and Morgan-Tian (arXiv:0809.4040v1 [math.DG], 2008). A version of Perelman's collapsing theorem states: "Let {M 3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) andi is closed or has possibly convex incompressible toral boundary. Then M 3 i must be a graph manifold for sufficiently large i". This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman's critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.The verification of Perelman's collapsing theorem is the last step of Perelman's proof of Thurston's geometrization conjecture on the classification of 3-manifolds. A version of the geometrization conjecture asserts that any closed 3-manifold admits a piecewise locally homogeneous metric. Our proof of Perelman's collapsing theorem is accessible to advanced graduate students and non-experts.