@article{BaerWafo2015, author = {B{\"a}r, Christian and Wafo, Roger Tagne}, title = {Initial value problems for wave equations on manifolds}, series = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, volume = {18}, journal = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {1385-0172}, doi = {10.1007/s11040-015-9176-7}, pages = {29}, year = {2015}, abstract = {We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Geometrically formal 4-manifolds with nonnegative sectional curvature}, series = {Communications in analysis and geometry}, volume = {23}, journal = {Communications in analysis and geometry}, number = {3}, publisher = {International Press of Boston}, address = {Somerville}, issn = {1019-8385}, pages = {479 -- 497}, year = {2015}, abstract = {A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S-4 or diffeomorphic to CP2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S-2 x S-2 holds in this class of manifolds.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes}, series = {Communications in mathematical physics}, volume = {333}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-014-2097-7}, pages = {1585 -- 1615}, year = {2015}, abstract = {Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.}, language = {en} }