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Published: 
December 17, 2002

Keywords: 
weighted L_{2} cohomology, locally symmetric space, Gaussian weight

Subject: 
53C, 22E, 14F

Abstract:

Let $(\mathcal{M},g)$ be a complete, noncompact Riemannian
manifold of finite volume. For $w:\mathcal{M} \rightarrow
(0,\infty)$ a weighting function, the $w$ weighted unreduced $L_2$
cohomology is defined as the usual unreduced $L_2$ cohomology
except that $\dvol$ is replaced by the measure $w \dvol$. This
paper proves that in the case $\mathcal{M} = \Gamma \bs G /K$ is a
locally symmetric space of nonpositive sectional curvature and
arbitrary rank whereupon $w(m) = \mbox{e}^{\mbox{dist}(m,p_0)^2}$
is the Gaussian relative to some basepoint $p_0$, the $w$ weighted
unreduced $L_2$ cohomology is isomorphic to the usual de Rham
cohomology. This isomorphism extends to the standard coefficient
bundles.
Note that weights for the de Rham cohomology of exponential decay
have already been constructed; see \cite{borel:83a},
\cite{borel:90a} and \cite{franke:98a}. The Gaussian weight
behaves differently in terms of coefficient bundles.

Acknowledgments:
This research supported by the U.M. VIGRE grant.
Author information:
1859 East Hall, University of Michigan, Ann Arbor, MI 481091109
stephnsb@umich.edu
http://www.math.lsa.umich.edu/~stephnsb/
 