Positive scalar curvature with skeleton singularities
We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (\(L^\infty \)) metrics that consolidate Gromov’s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles \(\le 2\pi \) 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 \(\le 2\pi \)) and arbitrary \(L^\infty \) isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities.
The authors would like to thank Rick Schoen, Brian White, Rafe Mazzeo, Pengzi Miao, and Or Hershkovits for stimulating conversations on the subject of this paper, as well as Gerhard Huisken, Dan Lee, André Neves, Yuguang Shi, and Peter Topping for their interest in this work. The first author would like to thank ETH-FIM for their hospitality, during which part of this work was carried out. The second author would like to thank the Ric Weiland Graduate Fellowship at Stanford, which partially supported the early portion of this research.
- 13.Li, C.: A polyhedron comparison theorem for 3-manifolds with positive scalar curvature. arXiv:1710.08067. Accessed 21 Dec 2017
- 16.Schoen, R.: Variational theory for the total scalar curvaturefunctional for Riemannian metrics and related topics, Topics incalculus of variations (Montecatini Terme, 1987), Lecture Notesin Math., vol. 1365, pp. 120–154, Springer, Berlin (1989)Google Scholar
- 18.Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities. arXiv:1704.05490. Accessed 17 Jan 2018
- 38.Sormani, C.: Scalar curvature and intrinsic flat convergence. https://www.degruyter.com/downloadpdf/books/9783110550832/9783110550832-008/9783110550832-008.pdf. Accessed 21 Dec 2017
- 43.Simon, L.: Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra (1983)Google Scholar
- 45.Mantoulidis, C.: Geometric variational problems in mathematical physics, Ph.D. thesis, Stanford University, (2017)Google Scholar