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

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Title: Inradius bounds for stable, minimal surfaces in 3-manifolds with positive scalar curvature
Author: Richardson, James
Degree Master of Science - MSc
Program Mathematics
Copyright Date: 2012
Publicly Available in cIRcle 2012-05-24
Abstract: 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.
URI: http://hdl.handle.net/2429/42368
Scholarly Level: Graduate

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