Search results for "Differential geometry"
showing 10 items of 462 documents
L2-torsion of hyperbolic manifolds
1998
The L^2-torsion is an invariant defined for compact L^2-acyclic manifolds of determinant class, for example odd dimensional hyperbolic manifolds. It was introduced by John Lott and Varghese Mathai and computed for hyperbolic manifolds in low dimensions. In this paper we show that the L^2-torsion of hyperbolic manifolds of arbitrary odd dimension does not vanish. This was conjectured by J. Lott and W. Lueck. Some concrete values are computed and an estimate of their growth with the dimension is given.
Wolfe's theorem for weakly differentiable cochains
2014
Abstract A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in R n with the space of flat m -cochains, that is, the dual space of flat chains of dimension m in R n . The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in R n . A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen–Koskela's concept of upper gradient of a function.
A rigidity problem on the round sphere
2015
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
Sets with constant normal in Carnot groups: properties and examples
2019
We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its sub-Riemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal directio…
Counting and equidistribution in Heisenberg groups
2014
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for …
Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian
2014
Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
2013
We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.
Tensor tomography on Cartan–Hadamard manifolds
2017
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. This work extends the results of Lehtonen (2016) to dimensions $n \geq 3$ and to the case of tensor fields of any order.
X-ray transforms in pseudo-Riemannian geometry
2016
We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\geq1$ and $n_2\geq2$ or vice versa and always when $n_1,n_2\geq2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space fro…
Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces
2014
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.