HARMONICS FORMS ON NON COMPACT MANIFOLDS

2L HARMONICS FORMS ON NON COMPACT MANIFOLDS.GILLES CARRONThe source of these notes is a series of lectures given at the CIMPA’s summerschool ”Recent Topics in Geometric Analysis”. I want to thank the organizers ofthis summer school : Ahmad El Soufi and Mehrdad Shahsahani and I also want tothank Mohsen Rahpeyma who solved many delicate problems.2Theses notes aimed to give an insight into some links between L cohomology,2L harmonics forms, the topology and the geometry of complete Riemannian man-ifolds. This is not a survey but a choice of few topics in a very large subject.2The first part can be regard as an introduction ; we define the space of L2harmonics forms, of L cohomology. We recall the theorems of Hodge and deRham on compact Riemannian manifolds. However the reader is assumed to befamiliar with the basic of Riemannian geometry and with Hodge theory.According to J. Roe ([55]) and following the classification of Von Neumann alge-2bra, we can classify problems on L harmonics forms in three types. The first one2(type I) is the case where the space of harmonics L forms has finite dimension,this situation is the nearest to the case of compact manifolds. The second (type2II) is the case where the space of harmonics L forms has infinite dimension butwhere we have a ”renormalized” dimension for instance when a discrete group actscocompactlybyisometryonthemanifold; agoodreferenceisthebookofW.Lueck([47]) and the seminal paper of M. Atiyah ([4]). The third type ...