A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.
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13 results for “L2-cohomology”
We give a topological interpretation of the space of L2-harmonic forms on finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The me…
This paper generalizes L2 cohomology theory for complex manifolds.
problem Developing a L2 cohomology theory for Hodge modules on infinite covering spaces.
method Formulating a conjectural generalization of L2-Mixed Hodge structures using Saito's Mixed Hodge Modules.
result Partial results in the conjectural generalization of L2-Mixed Hodge structures.
We compute the space of harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.
We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is …
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…
Smith theory, L2 cohomology, isometries of locally symmetric manifolds and moduli spaces of curvesmath.GT
We investigate periodic diffeomorphisms of non-compact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with non-vanishing Euler characteristic. In the i…
We describe compactifications of the moduli spaces of SU(2) monopoles on R3 as manifolds with corners, with respect to which the hyperKaehler metrics admit asymptotic expansions up to each boundary face. The boundary faces encode monopoles of charge k decomposing into widely separated monopoles of lower charge, and the…
The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended u…
For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Bau…
A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free …
Quiver varieties' geometry at infinity studied using Nakajima metric.
problem Understanding the geometry at infinity of quiver varieties.
method Using Melrose's approach to study the geometry at infinity of the Nakajima metric on reduced Hilbert schemes.
result Quiver varieties are quasi-asymptotically conical under generic conditions.
The paper bounds Betti numbers of complex-hyperbolic manifolds.
problem Estimating Betti numbers of complex-hyperbolic manifolds.
method Unitary holonomy, new monotonicity inequalities, peaking argument.
result Effective upper bounds for Betti numbers in various hyperbolic settings.