POSITIVE SCALAR CURVATURE: METHODS, RESULTS AND PERSPECTIVES

BERNHARD HANKE, THOMAS SCHICK AND OTHERS

1. Introduction and Overview: Definition of scalar curvature, Yamabe-problem, Kazdan-Warner trichotomy. (Hanke)

2. Obstructions I: Construction of Dirac-Operator on closed spin manifolds, index-obstruction against positive scalar curvature after Lichnerowicz and Hitchin (both following the exposition in Lawson-Michelsohn). (Pennig)

3. Obstructions II: Refined index obstruction with values in K_{∗}(C^{∗}π_{1}(M)): Short introduction of C^{∗}-algebras and their Ktheory, construction of an index problem, definition of assembly map via KK, Gromov-Lawson-Rosenberg conjecture and its connection with Baum-Connes, resp. Novikov conjecture. (Pennig, Norouzizadeh)

4. Obstructions III: Largeness as an obstruction to positive scalar curvature: Enlargeability and the codimension-2 obstruction (after Gromov-Lason) and unified treatment via coarse methods (after Hanke-Schick). (Schick)

5. Obstructions IV: Obstruction in dimension 4 by Seiberg-Witten theory. Schoen-Yau’s minimal hypersurfaces (Roman)

6. Constructions I: Gromov-Lawson’s surgery principle. (hanke)

7. Constructions II: Derivation of a bordism principle via handle decompositions and handle cancelations. Gromov-Lawson’s theorem on the existence of positive scalar curvature on closed simply connected non-spin manifolds of dimension at least 5. (Thom)

8. Constructions III: Stolz’ theorem on the classification of closed simply connected spin manifolds of dimension at least 5 and admitting metrics of positive scalar curvature. The Baum-Connes conjecture implies the stable Gromov-Lawson-Rosenberg conjecture (following Stolz’ contribution in the 2001 Trieste proceedings).

9. Families obstructions: The space of positive scalar curvature metrics, concordance, Stolz’ R_{n}-groups, Rubermann’s construction of concordant, nonhomotopic positive scalar curvature metrics in dimension 4. (Mitchener)