2

JOHN ROE

any complete, simply-connected, nonpositively curved manifold is u;-large.

(1.1) THEOREM: (See theorem 6.16,) Let (M,go) be a complete simply-connected

Riemannian manifold of non-positive curvature. Then there is no metric g\ on

M with /i A7 o for some positive constant A and such that the scalar curvature

of gi is uniformly positive.

In the form stated, this is a result proved by Gromov and Lawson [36], and

it has as a corollary their famous result that no compact manifold that has

a metric of non-positive sectional curvature can also have a metric of positive

scalar curvature (see also [61, 62], [35, 34], [45]). But it appears that the full

class of u;-large manifolds is larger than the corresponding class of hyperspherical

manifolds in their work.

(1.2) THEOREM: (See theorem 6.24^ Let (M, go) be a complete simply-connected

Riemannian 2n-manifold of non-positive curvature, and let g\ be another metric

on M with g\ A/o for some positive constant A. Then the essential spectrum

of the Laplacian on square integrable n-forms on (M,g\) contains zero.

This does not seem to be known even for #0 = h, except for constant or

nearly constant negative curvature. It could be regarded as a weak form of

a conjecture of Singer, that for strictly negative curvature there is an infinite-

dimensional space of L2 harmonic n-forms (see [24]). The proof uses the index

theorem applied to the signature operator.

(1.3) THEOREM: (See theorem 6.19,) Let X be a complete simply-connected

Kdhler manifold of non-positive (sectional) curvature, and let Y be any Kdhler

manifold admitting a proper holomorphic map to X that increases distances by

no more than a bounded amount. (Y need not have the same dimension as X.)

Let L be a holomorphic line bundle over Y equipped with a metric of uniformly

positive curvature. Then for any holomorphic vector bundle E over Y equipped

with a metric of bounded curvature, there is a constant /i 0 such that the

bundle L** 0 E has an infinite-dimensional space of L2 holomorphic sections.

This is a non-compact version of Cartan's 'Theorem B' [31, page 159].

As a final application, Weinberger [69] has outlined a proof by means of the

index theorem of this paper of Novikov's theorem [51] on the topological invari-

ance of the rational Pontrjagin classes of a compact manifold. The non-compact

manifold to which the index theorem is applied is a tubular neighbourhood of a

submanifold of the original compact manifold.

Here is a more detailed overview of the paper. Chapter 2 begins by introducing

appropriate categories on which coarse cohomology will be functorial. The mor-

phisms in these categories are referred to as 'homologous' maps; they are proper

maps with a uniform control on their expansiveness. Coarse cohomology (de-

noted HX*) is then defined as the cohomology of a modified Alexander-Spanier

complex: if M is a metric space of an appropriate kind, then the j-cochains

of coarse cohomology are functions on Mq+1 whose support is compact in each