The η-invariant has been defined for C^∞-manifolds by M.F. Atiyah, V.K. Patodi and I.M. Singer, and more recently for manifolds with corners by A. Hassell, R. Mazzeo and R.B. Melrose, and for stratified PL manifolds by H. Moscovici and F.B. Wu. In the present work, this invariant is generalized in the framework of lipschitz riemannian manifolds. This involves selfadjoint extensions of the signature operator on a lipschitz manifold with boundary, and measurable differential forms which represent the Pontryagyn classes of the manifold. This allows us to extend from smooth to topological manifolds the Atiyah-Patodi-Singer index theorem for flat bundles.

In this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds, a class of manifolds generalizing Euclidian n-space. Subject to the basic assumption of analyticity near infinity, we show that resolvent of the Laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum. This involves a geometric interpretation of the complex scaling method in terms of deformations in the Grauert tube of the manifold. We then show that the number of resonances (poles of the meromorphic continuation of the resolvent) in a conic neighborhood of $\mathbb{R}_+$of absolute value less than $r^2$ is $\mathcal O(r^n)$. Under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow, we prove a finer, Weyl-type upper bound for the counting function for resonances in small neighborhoods of the real axis. This estimate has an exponent which involves the dimension of the trapped set of the geodesic flow.

We present complete classification results for tight contact structures on two classes of 3-manifolds: (i) torus bundles which fiber over the circle and (ii) circle bundles which fiber over closed surfaces.

We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ≥ 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the H^s Hilbert-class topologies for s > (n/2) + 1, these subgroups are C^∞ differential manifolds. We consider these new diffeomorphism groups with an H¹-equivalent right invariant metric, and prove the existence of unique smooth geodesics η(t, ·) of this metric, as well as existence and uniqueness of the Jacobi equations associated to this metric. Geodesics on these subgroups are, in fact, the flows of a time-dependent velocity vector field u(t, x), so that ∂_tη(t, ·) = u(t, η(t, ·)) with η(0, x) = x, and remarkably the vector field u(t, x) solves the so-called Lagrangian averaged Euler (LAE-α) equations on M. These equations, and their viscous counterparts, the Lagrangian averaged Navier-Stokes (LANS-α) equations, model the motion of a fluid at scales larger than an a priori fixed parameter α > 0, while averaging (or filtering-out) the small scale motion, and this is achieved without the use of artificial viscosity. We prove that for divergence-free initial data satisfying u = 0 on ∂M, the LAE-α equations are well-posed, globally when n = 2. We also find the boundary conditions that make the LANS-α equations well-posed, globally when n = 3, and prove that solutions of the LANS-α equations converge when n = 2, 3 for almost all t in some fixed time interval (0, T) in H^s, s ∊ (n/2 + 1, 3) to solutions of the LAE-α equations, thus confirming the scaling arguments of Barenblatt & Chorin.