Non-smooth Lorentzian and symplectic geometry

Our research project addresses global problems of analysis on manifolds with non-smooth Lorentzian or symplectic structures. These settings are the fundamental geometries underlying mathematical relativity theory and an invariant analysis of normally hyperbolic partial differential operators. If the basic tensor fields defining the Lorentzian or symplectic structure cease to be sufficiently smooth, many of the classical methods fail and key results, such as local uniqueness and existence of geodesics, are no longer true. However, low regularity of the Lorentzian metric is a relevant aspect in models of impulsive gravitational waves or cosmic strings, and the need for a generalized symplectic geometry arises in Hamiltonian mechanics, e.g., in case of rough potential energies, in microlocal analysis of differential operators with non-snooth coefficients, or when describing the geodesic flow on a non-smooth Pseudo-Riemannian manifold in terms of symplectomorphisms. We will apply methods from the global analysis of distributions and from geometric nonlinear theories of generalized functions which are based on asymptotic estimates of regularizing sequences. Our focus in the context of generalized Lorentzian geometry is on the Cauchy problem for wave equations and causality properties on non-smooth curved space-times with emphasis on an extension of the theory of global hyperbolicity, in particular including a generalized theory of Cauchy hypersurfaces. We will study the specific features of causality notions, such as time-orientation, causal future and past, domain of dependence, and Cauchy developments, in relation with properties of non-smooth global hyperbolicity. This will lead to an advanced analysis of global solutions to wave equations on non-smooth curved space-times in terms of support or microlocal properties in dependence on the initial data and the operator symbol. As for generalized symplectic geometry, we first have to develop an extension of symplectic linear algebra to the case of modules over the ring of generalized numbers to prepare for appropriate tangent space constructions. In particular, the notion of generalized Lagrangian subspace has to be defined and investigated. Based on these, we will proceed to build a basic theory of non-smooth symplectic forms on manifolds and establish a generalization of Darboux`s theorem, i.e., generalized symplectomorphisms with the standard cotangent bundle. Furthermore, we will study generalized Lagrangian submanifolds and the possibilities to represent them by means of generating functions as well as their interplay with generalized symplectomorphisms. In addition, a theory for non-smooth Hamiltonian vector fields and flows will be developed. The results will be applied to wave front sets of kernels of generalized Fourier integral operators and to the bicharacteristc flow associated with the wave operator on a non-smooth curved space-time. More specifically, we will investigate their precise interrelation with geodesics and with wave front sets of generalized solutions.

The research project addresses global problems of analysis and geometry on manifolds with non-smooth Lorentzian or symplectic structures. These settings are the fundamental geometries underlying mathematical relativity theory and for Hamiltonian nonlinear mechanics. Under realistic conditions of low regularity of the basic objects, many of the classical methods fail and key results are no longer true, requiring new derivations in qualtitatively very different settings. This has been achieved in course of the project, in particular, in research on impulsive gravitational waves, cosmic strings, singularity theorems of general relativity, and in the form of an extended theory of causality.Furthermore, in a newly developed generalized symplectic geometry, upon establishing new appropriate constructions in the tangent spaces, the theoretical foundations of nonsmooth symplectic forms on manifolds have been laid out and a generalization of the central theorem by Darboux has been proved, which opened the way to nonsmooth Hamiltonian mechanics.