Maximum curvature for curves in manifolds of sectional curvature at most zero or one
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- by Ben Andrews and Changwei Xiong PDF
- Proc. Amer. Math. Soc. 147 (2019), 5403-5416 Request permission
Abstract:
We prove a sharp lower bound for the maximum curvature of a closed curve in a complete, simply connected Riemannian manifold of sectional curvature at most zero or one. When the bound is attained, we get the rigidity result. The proof utilizes the maximum principle for a suitable two-point function. In the same spirit, we also obtain a lower bound for the maximum curvature of a curve in the same ambient manifolds which has the same endpoints with a fixed geodesic segment and has a prescribed contact angle. As a corollary, the latter result applies to a curve with free boundary in geodesic balls of Euclidean space and hemisphere.References
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Additional Information
- Ben Andrews
- Affiliation: Mathematical Sciences Institute, Australian National University, ACT 2601, Australia
- MR Author ID: 317229
- ORCID: 0000-0002-6507-0347
- Email: Ben.Andrews@anu.edu.au
- Changwei Xiong
- Affiliation: Mathematical Sciences Institute, Australian National University, ACT 2601, Australia
- MR Author ID: 1049017
- Email: changwei.xiong@anu.edu.au
- Received by editor(s): March 12, 2019
- Published electronically: July 30, 2019
- Additional Notes: This research was partly supported by Discovery Projects grant DP120102462 and Australian Laureate Fellowship FL150100126 of the Australian Research Council.
- Communicated by: Jia-Ping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5403-5416
- MSC (2010): Primary 53C20, 58J60, 53A04
- DOI: https://doi.org/10.1090/proc/14708
- MathSciNet review: 4021098