Positive scalar curvature and minimal hypersurfaces

Positive Scalar Curvature and Minimal Hypersurfaces

We show that the minimal hypersurface method of Schoen and Yau can be used for the “quantitative” study of positive scalar curvature. More precisely, we show that if a manifold admits a metric g with sg ≥ |T | or sg ≥ |W |, where sg is the scalar curvature of of g, T any 2-tensor on M and W the Weyl tensor of g, then any closed orientable stable minimal (totally geodesic in the second case) hyp...

06 Singular Minimal Hypersurfaces and Scalar Curvature

Finding obstructions to positive scalar curvature and getting structural insight is presently based on two competing approaches: one path which is most travelled works in the context of spin geometry and gives quite a direct link to topology (cf. [GL1-2] and [G]). The second, much less used but a priori more general method of attack analyzes minimal hypersurfaces within the manifold under consi...

Hypersurfaces with Constant Scalar Curvature

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of com...

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

Positive Scalar Curvature and Surgery