Singularity and comparison theorems for metrics with positive scalar curvature
- Chao Li.
- [Stanford, California] : [Stanford University], 2018.
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- 1 online resource.
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- In this thesis, we discuss metric properties of positive scalar curvature. Metrics with positive scalar curvature naturally arise from various geometric and physical problems. However, some basic questions of positive scalar curvature were unknown. Specifically, can one conclude that the scalar curvature of a metric is positive, just based on measurement by the metric, without taking any derivative? Such questions are usually answered via geometric comparison theorems. They are also built upon a good understanding of the singular set, along which a sequence of metrics with uniformly bounded curvature degenerate. The primary contributions of this thesis are twofold: Firstly, we study the effect of uniform Euclidean singularities on the Yamabe type of a closed, boundaryless manifold. We show that, in all dimensions, edge singularities with cone angles ≤ 2π along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1- skeletons, exhibiting edge singularities (angles ≤ 2π) and arbitrary L∞ isolated point singularities. Secondly, we establish a geometric comparison theorem for 3-manifolds with positive scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove that the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron. From the viewpoint of metric geometry, our results show that R ≥ 0 is faithfully captured by polyhedra. They suggest the study of "R ≥ 0" with weak regularity assumptions, and the limit space of manifolds with scalar curvature lower bounds.
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- Submitted to the Department of Mathematics.
- Thesis Ph.D. Stanford University 2018.
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