1,121 research outputs found
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
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
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
is an absolutely continuous measure on S, then we compute the tangent
cone of P(M) at .Comment: final version, 11 pages. Part of an earlier version is split off to
arXiv:1701.0229
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
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
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
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