Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

DOI10.1515/CRELLE-2021-0060arXiv2012.04430MaRDI QIDQ6355574

Publication date: 8 December 2020

Abstract: In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

Mathematics Subject Classification ID

Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) (58J60) PDEs on manifolds (35R01) Ricci flows (53E20)

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