Invariant Currents on Limit Sets

We relate the L^2 cohomology of a complete hyperbolic manifold to the invariant currents on its limit set.Comment: 27 page

On the spectrum of a finite-volume negatively-curved manifold

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to -1 and with an infinite number of gaps in the spectrum of the function Laplacian.Comment: 17 pages, statement of Theorem 2 improve

On tangent cones in Wasserstein space

If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $\mu$ is an absolutely continuous measure on S, then we compute the tangent cone of P(M) at $\mu$.Comment: final version, 11 pages. Part of an earlier version is split off to arXiv:1701.0229

Remark about scalar curvature and Riemannian submersions

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning the scalar curvature of a smooth limit space arising in a bounded curvature collapse.Comment: final versio

The collapsing geometry of almost Ricci-flat 4-manifolds

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.Comment: final final versio

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.Comment: final version, 28 page

A Dolbeault-Hilbert complex for a variety with isolated singular points

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.Comment: final versio

Secondary analytic indices

We define analytic indices which involve the eta form and the analytic torsion form. We show that these indices are independent of the geometric choices made in their definitions, and hence are topological in nature.Comment: 75 pages, uuencoded, compressed dvi fil

Collapsing and the Differential Form Laplacian : The Case of a Singular Limit Space

In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter.Comment: 25 page

Some geometric calculations on Wasserstein space

We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.Comment: final versio