Ricci flow on manifolds with boundary with arbitrary initial metric
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Tsz-Kiu Aaron Chow
Abstract
In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed 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.
Acknowledgements
The author would like to express his gratitude to his advisor Professor Simon Brendle for his continuing support, his guidance and many inspiring discussions. The author would also like to thank the anonymous referee for his/her very careful reading and insightful comments, and for his/her valuable suggestions to improve the exposition of the previous version.
References
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Articles in the same Issue
- Frontmatter
- An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field
- Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)
- Prescribing Ricci curvature on homogeneous spaces
- Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
- Ricci flow on manifolds with boundary with arbitrary initial metric
- Unbounded negativity on rational surfaces in positive characteristic
- Extending torsors on the punctured Spec(Ainf)
- Non-isomorphic 2-groups with isomorphic modular group algebras
- Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)
