Gaussian weighted unreduced L₂ cohomology of locally symmetric spaces

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.

This research supported by the U.M. VIGRE grant.

1859 East Hall, University of Michigan, Ann Arbor, MI 48109-1109
stephnsb@umich.edu
http://www.math.lsa.umich.edu/~stephnsb/