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 , 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], ). 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 ). 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  has outlined a proof by means of the
index theorem of this paper of Novikov's theorem  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