Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion me...

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in π1M , and that the double cosets for crossing surfaces are also separable. We deduce that if there is a ‘sufficient’ collection of surfaces in M , then π1M is virtually the fundamental group of a special non-positively curved cube complex. We provide a sufficient collection for ...

This theorem is false if e ^ 0. Indeed, on the unit disc, the Kâhler metric (1 — zz)~dzdz (where n is any integer ^ 3) is complete and its curvature function K satisfies K < 0 and \K(z)\ S C(d(0, z))~. (0 = origin ofC.) The next theorem and its corollary provide information about the absence of holomorphic p-forms (p ^ 1) when the manifold is positively curved. For compact M, the result was kno...

Let X be an n-dimensional compact complex manifold, L → X a positive holomorphic line bundle, and H the space of positively curved hermitian metrics on L. The purpose of this article is to prove that geodesics in the infinite-dimensional symmetric space H can be uniformly approximated by geodesics in the finite-dimensional symmetric spaces Hk = GL(Nk + 1,C)/U(Nk + 1,C), where Nk + 1 = dim(H(L))...

Let Γ be a discrete group, defined by a presentation P = 〈ai | rj〉, say, or as the fundamental group of a connected CW-complex X. Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell E i for each ai (oriented accordingly), and one 2-cell E 2 j for each rj , with attaching map ∂E j → X (1) P that reads off the word rj in the generators {ai}. Then, by the Seifert–van Kampen Th...

In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompac...