for this workshop
Analysis and geometry on pseudohermitian manifolds
American Institute of Mathematics, San Jose, California
Sorin Dragomir, Howard Jacobowitz, and Paul Yang
This workshop, sponsored by AIM and the NSF, will be devoted to mathematical analysis and differential geometry on pseudohermitian manifolds. The analysis of solutions to the tangential Cauchy-Riemann equations is best performed in the presence of nondegeneracy assumptions on the given CR structure. For then a contact form $\theta$ may be chosen and differential geometric methods become available, for instance to compute CR invariants (e.g. the Chern-Moser tensor, or Kohn-Rossi cohomology) in terms of pseudohermtian invariants. An array of differential geometric objects (e.g. a sub-Riemannian structure along the maximally complex distribution together with the resulting Carnot-Caratheodory distance function, the Webster metric, the Tanaka-Webster connection embedding the subRiemannian geometry at hand into connection theory, the Fefferman metric bringing into the picture methods and results from Lorentzian geometry) spring naturally from $\theta$ and manifest either as sources of problems (e.g. the CR Yamabe problem, the existence and partial regularity problem for subelliptic harmonic maps from a CR manifold) or as geometric tools (e.g. the description of Cartan chains as projections of null geodesics of Fefferman's metric).
The workshop will concentrate on the following themes:
- Applying subelliptic theory to pseudohermitian geometry, as prompted by the occurrence, on any strictly pseudoconvex pseudohermtian manifold $(M,\theta)$, of the sublaplacian $\Delta_b$ and of the Kohn operator $\Box_b$, and exploiting the interrelation between hyperbolic and subelliptic theories [as urged by the presence of the Fefferman metric, a Lorentzian metric on the total space of the canonical circle bundle over $(M,\theta)$]. In particular one aims for a version of Kohn-Hodge-de Rham theory on CR Lie groups (endowed with left invariant pseudohermitian structures).
- Investigating the relationship between pseudohermitian geometry and spacetime physics, in relation to the occurrence of shear free null geodesic congruences on certain Lorentzian manifolds (e.g. Fefferman like spacetimes, Godel's universe) leading to orbit spaces admitting naturally defined strictly pseudoconvex CR structures. In particular, the CR embedding problem as related to Einstein's gravitational field equations.
- A new curvature invariant called the $Q$-prime curvature associated to a CR structure was recently developed together with intrinsic linear operators (the $P$-prime operators). These are the natural replacement for the $Q$-curvature and the Paneitz operator for CR geometry. These curvature invariants give rise to global CR invariants prompting new investigations in relation to global topological invariants.
- analysis in several complex variables,
- theory of Hormander systems of vector fields and subelliptic partial differential equations,
- differential geometry on CR, pseudohermitian, and Lorentzian manifolds.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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