拟纤维化边界度量的$L^2$上同调

A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

A manifold with fibered cusp metrics $X$ can be considered as a geometrical generalization of locally symmetric spaces of $\mathbb{Q}$-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology $H^p(X)$. Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge-Laplace-Operator on $Ω^p(X)$.

An index formula on manifolds with fibered cusp ends

We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dir…

具有纤维化角的流形乘积

Products of manifolds with fibered corners

本文定义了具有纤维化角的流形范畴及其乘积与横向纤维积，提出有序乘积作为笛卡尔积的爆破分辨，并证明其是楔形度量和拟纤维边界度量的自然乘积。

拟纤维化边界伪微分算子

Quasi-fibered boundary pseudodifferential operators

本文为拟纤维化边界度量发展伪微分演算，引入完全椭圆QFB算子并证明其Fredholm性质，进而构造Hodge-deRham算子的拟基本解，证明L²调和形式空间有限维且具有更快的无穷远衰减。

Cup-products in L q,p -cohomology: discretization and quasi-isometry invariance

We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomology of bounded geometry contact manifolds.

The cohomology of a variation of polarized Hodge structures over a quasi-compact Kähler manifold

A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.