A metric approach to the study of manifolds of positive scalar curvature

Overview & Synopsis

This thesis is dedicated to the topology and the geometry of Riemannian manifolds of positive scalar curvature. To explore the structure of such manifolds, we adopt a metric perspective, specifically through two metric generalisations of the notion of positive scalar curvature.First, we focus on the topology of 3-manifolds of positive scalar curvature, and we provide a new obstruction to the existence of complete Riemannian metrics of positive scalar curvature on non-compact 3-manifolds. More precisely, we prove that if an orientable 3-manifold M admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and ��^2 x ��^1 summands. As a consequence, the manifold M admits a complete metric of uniformly positive scalar curvature. This result constitutes a generalisation of a theorem by Gromov and Wang, and its proof builds upon a different approach of metric and topological nature. More generally, the topological decomposition holds without any assumption on the scalar curvature, relying instead on a metric estimate on the filling discs of closed curves in the universal cover, based in the notion of fill radius. Moreover, the decay rate in the decomposition theorem is optimal, since the manifold ℝ^2 x ��^1 admits a complete metric of positive scalar curvature decaying exactly quadratically at infinity, yet it does not decompose as a connected sum of spherical manifolds and ��^2 x ��^1 summands.Then, we explore the notion of macroscopic scalar curvature and its relation to the systolic geometry of a manifold. More precisely, we derive a macroscopic version of a celebrated systolic inequality by Bray-Brendle-Neves on the area of non-contractible 2-spheres in a manifold of positive scalar curvature. We show that if a complete Riemannian n-manifold with non-trivial codimension 1 homology with ℤ_2-coefficients or ℤ-coefficients has positive macroscopic scalar curvature, then it admits a non-nullhomologous hypersurface of small Urysohn (n-2)-widht. The proof of this result is based on an adaptation of Guth's macroscopic version of the Schoen-Yau descent argument.