On a compact connected spin manifold the index theorem by Atiyah and Singer gives lower bounds for the dimension of the kernel of the Dirac operator. Metrics for which the lower bound is attained are called D-minimal. It is conjectured that generic metrics are D-minimal and that non-D-minimal metrics exist on any manifold of dimension at least 3. These conjectures go back to Hitchin’s article « Harmonic spinors » where first steps in this program were done, and they were clearly conjectured by C. Bär and M. Dahl. That D-minimal metrics are generic was proven by Bär, Dahl, Humbert and myself using Gromov-Lawson type surgery methods and bordism theory. The existence of non-D-minimal metrics in dimension at least 7 is current work with Bunke, Pilca and Nowaczyk and uses recent progress by Crowley and Schick about the topology of the space of metrics with positive scalar curvature.

Résumé

http://www.math.univ-metz.fr/~aga/