Anisotropic deformation on contact three-manifolds

Abstract

Anisotropic deformation, sometimes generically called adiabatic deformation, is a metric blow-up that is natural to construct on contact manifolds. Such manifolds admit a Riemannian metric which is adapted to the contact distribution in the following sense: the metric $g$ consists of a horizontal part $g_h$, that comes from a subRiemannian structure on the tangent distribution, and another one $g_v$ on the vertical direction. The deformation then rescales the metric by a factor of $1/\epsilon^2$ on the vertical component. Every variation of the value of $\epsilon$, especially in the limit $\epsilon\to 0$, massively affects almost every metric-dependent attribute of the manifold. Gromov proved that the induced metric space converges in the Gromov-Hausdorff sense to the subRiemannian one. This thesis is a collection of recent results and possible future developments in the study of anisotropic limits. It starts with a description of some geometrical features of the anisotropic limit, namely the convergence of some Riemannian geodesics to Reeb orbits through \textit{spiraling} subRiemannian geodesics. Then the discussion continues with the study of the perturbation of some Laplace and Dirac-type operators. The purpose is to analyze the convergence of such operators in the resolvent sense.