Cohomologie $L^2$ des variétés QALE

Sur la L2-cohomologie des varietes a courbure negative

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 method consists first in comparing L2-cohomology with weighted L2-cohomology thanks to previuos works done by T. Ohsawa, and then in identifying these weighted spaces.

Cohomologie feuilletée du flot affine de Reeb sur la variété de Hopf ${\Bbb S}^n\times {\Bbb S}^1$

We determine explicitly the foliated cohomology $H_{\cal F}^\ast (M)$ of the affine Reeb flow ${\cal F}$ on the Hopf manifold ${\Bbb S}^n\times {\Bbb S}^1$. The vector space $H_{\cal F}^1(M)$ contains exactly the obstructions to solve the cohomological equation $X\cdot f=g$ where $f$ and $g$ are $C^\infty $-functions and $X$ is any non singular vector field defining the foliation ${\cal F}$. The topological dual of $H_{\cal F}^1(M)$ is the space of distributions invariant by $X$.

Hodge Theory and Symplectic Boundary Conditions

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various form decomposition and also isomorphisms between the symplectic cohomologies and the spaces of harmonic fields. These novel boundary conditions can be applied in certain cases to study relative symplectic cohomologies and Lefschetz maps between relative de Rham cohomologies. As an …

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。