Weighted \(L_2\) cohomology of asymptotically hyperbolic manifolds (Q5928529)

The author considers a Riemannian manifold \(M\) (\(\dim M = n\)) with finitely many (say, \(m\)) hyperbolic ends \({\mathcal E}_i = (0, \infty)\times S^{n-1}\), a vector \(\Xi = (\Xi (1), \dots , \Xi (m))\in\mathbb R^m\), a corresponding weighting function \(\omega _\Xi\) which restricts to \(\exp ((2\Xi (i) + n-1)t)\) on each \({\mathcal E}_i\), and the complex of all the forms which are (together with their exterior differentials) \(L_2\)-integrable with respect to the volume element \(\omega_\Xi\cdot dvol\), \(dvol\) being the Riemannian volume element on \(M\). The corresponding weighted cohomology \(H^\ast _{2,\Xi}(M)\) is the main object of interest here. Among other results, the author shows that the weighted cohomologies are of finite dimension unless some of \(\Xi (i)\)'s belong to the set \(\{ -n+1, -n+2, \dots, -1, 0\}\).