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Maximum curvature for curves in manifolds of sectional curvature at most zero or one


Authors: Ben Andrews and Changwei Xiong
Journal: Proc. Amer. Math. Soc. 147 (2019), 5403-5416
MSC (2010): Primary 53C20, 58J60, 53A04
DOI: https://doi.org/10.1090/proc/14708
Published electronically: July 30, 2019
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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.


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Additional Information

Ben Andrews
Affiliation: Mathematical Sciences Institute, Australian National University, ACT 2601, Australia
Email: Ben.Andrews@anu.edu.au

Changwei Xiong
Affiliation: Mathematical Sciences Institute, Australian National University, ACT 2601, Australia
Email: changwei.xiong@anu.edu.au

DOI: https://doi.org/10.1090/proc/14708
Keywords: Geodesic curvature, Riemannian manifold, free boundary
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
Article copyright: © Copyright 2019 American Mathematical Society