Stephen S. Bullock
Weighted L2 Cohomology of Asymptotically Hyperbolic Manifolds
||March 19, 2001
||weighted cohomology, weighted L2 cohomology, weighted Hodge theory
||53C, 14F, 53A, 57T
The main results are summarized by Figure 1.
They demonstrate the resiliency of the isomorphism constructed in
[Nai99] between weighted
cohomology and a variant of weighted L2 cohomology. Our attention
is restricted from generic locally symmetric
spaces to spaces whose ends are hyperbolic,
(0,∞)×(S1)n-1, and carry exponentially warped product
metrics. For weighting functions which are exponential in the Busemann
coordinates of these ends, the standard w weighted L2 cohomology will
be utilized in lieu of the variant defined in [Fra98]. The
resulting standard w weighted L2 cohomology groups may
be infinite dimensional vector spaces,
but the precise weighting functions at which this undesirable behavior
occurs are characterized. For the remaining exponential
weights, the w weighted L2 cohomology is again an analogue of weighted
cohomology. An immediate consequence of finite dimensionality of the
standard w weighted L2 groups is a w weighted Hodge theory summarized
by a strong w weighted Kodaira decomposition. This is outlined in the
After the asymptotically hyperbolic case is complete, the literature
on weighted Hardy inequalities on the half line is used to derive
certain extensions to some non-hyperbolic end metrics and non-exponential
weighting functions. The two most immediate applications are as follows.
First, say a function on the half line k(t) satisfies
k' k-1 ≦ -c for c>0. Then one may replace the exponential
in the metric of (0,∞)×(S1)n-1 by k(t) and weight by powers
of k(t) rather than e-t, and Figure 1 holds.
Second, the analysis allows one to
consider weighting functions which on each end are
w(t) = eαt2 for α ∈ R. These weighting functions
compute either de Rham cohomology or compactly supported de Rham cohomology
when α < 0 or α > 0, respectively.
This research was partially supported by the Clay mathematics institute liftoff program.
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