Inradius bounds for stable, minimal surfaces in 3-manifolds with positive scalar curvature

• Main Author: Richardson, James
• Language: English
• Published: University of British Columbia 2012
• Online Access: http://hdl.handle.net/2429/42368

Concrete topological properties of a manifold can be found by examining its geometry. Theorem 17 of his thesis, due to Myers [Mye41], is one such example of this; it gives an upper bound on the length of any minimizing geodesic in a manifold N in terms of a lower positive bound on the Ricci curvature of N, and concludes that N is compact. Our main result, Theorem 40, is of the same flavour as this, but we are instead concerned with stable, minimal surfaces in manifolds of positive scalar curvature. This result is a version of Proposition 1 in the paper of Schoen and Yau [SY83], written in the context of Riemannian geometry. It states: a stable, minimal 2-submanifold of a 3-manifold whose scalar curvature is bounded below by κ > 0 has a inradius bound of ≤√(8/3) π/√κ, and in particular is compact. === Science, Faculty of === Mathematics, Department of === Graduate