Witten’s perturbation and lefschetz formula on singular spaces

Resumo

The generalization of several topics of golbal analysis for compact or complete riemannian manifolds to the case of stratum with adapted metrics, which are not compact neither complete, was started by cheeger around 1980. Currently it is in a gather moment because of the contributions of other important mathematicians [1-3,8-12]. In particular, the director of this phd thesis was coauthor of an analytical version of morse inequalities in stratum, equipped with adapted metrics [3]. Specifically, in that context, a version of witten deformation for the de rham complex induced by a morse function was studied. In that case, the "morse functions" used may present "critical points" in the metric completion of the manifold, because it is not compact neither complete. For the same reason, several asociated hilbert complexes could appear, generating a huge analytical complexity. For compact manifolds, witten also considered the complex of eigenforms asociated to "small" eigenvalues when the parameter of the witten´s complex tends to infinity, which is defined by the critical points of the morse function and the orbits between them. For compact manifolds, this programme was developed independently by bismut-zhang and helfer-sjöstrand [4,7]. For stratum with adapted metrics , the only generalization in a very restrictive case was made by ludwig [12], remaining open for arbitrary stratum with general adapted metrics. Another open problems in this area are the generalization for stratum of the analytical version of leftchetz´s formula and the index and hopf theorems for spin structures. The initial hipothesis is that the generalizations indicated previously must be true. Bibliography: 1. P. Albin, e. Leichtnam, r. Mazzeo and p. Piazza, hodge theory on cheeger spaces, j. Reine angew. Math. 744 (2018), 29--102. 2. P. Albin, e. Leichtnam, r. Mazzeo and p. Piazza