The PI proposes to study differential-geometric aspects of rigidity: which groups can act on a compact manifold preserving a rigid geometric structure of given type? When a given group acts, what can be the geometric and topological properties of the underlying manifold? The PI's work focuses on these questions in the setting of pseudo-Riemannian metrics and conformal structures.
The proposed research draws from three active branches of mathematics: differential geometry, dynamics, and Lie groups. The objects of study, pseudo-Riemannian and, in particular, Lorentzian metrics, arise in general relativity. The spacetimes of interest to relativists are noncompact, but they often consider conformal compactifications. Isometric or conformal automorphisms are important in the search for models of spacetime, because they simplify the background partial differential equations, and they correspond to conservation laws.
For this project the PI studied questions in the following topics: First, the PI studied isometries of Lorentzian manifolds, which are the setting for general relativity. Symmetry is a very important feature of spacetime models. Boundaries of Lorentzian manifolds and other semi-Riemannian manifolds typically have a conformal structure. With Charles Frances of the University of Orsay, the PI found a precise local description of conformal flows on analytic Lorentzian manifolds, and they worked on the Lorentzian Lichnerowicz Conjecture. Parabolic geometries are a generalization of conformal structures. In separate projects with Andreas Cap of the University of Vienna and Katharina Neusser of Australian National University, the PI has begun developing a general theory for flows on such geometries. Last, the PI has studied a significant question, called the Smooth Cannon Conjecture, about boundaries of delta-hyperbolic groups, which are algebraic objects endowed with some geometry resembling hyperbolic space. Significant broader impacts of this project were: 1) PI co-organized "Workshop on Cartan connections, homogeneous spaces, and dynamics," at the Erwin Schroedinger Insitute for Mathematical Physics in July 2011. 2) PI founded writing workshop for graduate students at the University of Maryland, and ran this in each spring 2011, 2012, and 2013. These will continue into PI's next project, starting with fall 2014. 3) PI collaborated with young researchers, graduate student Wouter van Limbeek from the University of Chicago and post-doc Katharina Neusser from Australian National University. This project supported visits by each of them to the University of Maryland, during which they gave talks in our Geometry-Topology seminar. 4) PI supported and mentored University of Maryland graduate student Jean-Philippe Burelle in summer research project in 2013 on 3-dimensional Lorentzian metrics with constant scalar invariants. 5) PI gave numerous talks during the grant period, including colloquia at Howard University, University of Chicago, University of Wisconsin, University of Illinois at Chicago, and Australian National University. The PI gave a plenary lecture at the workshop on "Discrete Groups and Geometric Structures, with Applications" in Oostende, Belgium, in June 2011. The PI was the Evans Lecturer at Cornell in January 2013. The PI developed and delivered mini-courses at the GEAR Junior Retreat in July - August 2012 and at the Srni Winter School in Geometry and Physics in January 2014. The following articles were accepted and published during the funding period: C. Frances and K. Melnick, "Formes normales pour les champs conformes pseudo-riemanniens" ["Normal forms for conformal pseudo-Riemannian vector fields"], Bulletin de la Societe Mathematique de France 141 no. 3 (2013) 377-421. A. Cap and K. Melnick, "Essential Killing fields of parabolic geometries: conformal and projective structures" Central European Journal of Mathematics 11 no. 12 (2013) 2053-2061. A. Cap and K. Melnick, "Essential Killing fields of parabolic geometries" Indiana University Mathematics Journal 62 no. 6 (2013) 1917-1953. The PI has also posted the following article on the arXiv, to be submitted this week: S. Dumitrescu and K. Melnick, "Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist" arXiv:1406.2302. Last, the PI's article with Katharina Neusser, tentatively titled "Strongly essential automorphisms and curvature vanishing on parabolic geometries" is close to completion.
