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

Andrews, Ben; Xiong, Changwei

doi:10.1090/proc/14708

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

F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451–547. MR 855173, DOI 10.1512/iumj.1986.35.35028
Ben Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012), no. 3, 1413–1418. MR 2967056, DOI 10.2140/gt.2012.16.1413
Ben Andrews, Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, Int. Press, Somerville, MA, 2015, pp. 1–47. MR 3381494, DOI 10.4310/SDG.2014.v19.n1.a1
Ben Andrews and Haizhong Li, Embedded constant mean curvature tori in the three-sphere, J. Differential Geom. 99 (2015), no. 2, 169–189. MR 3302037
Simon Brendle, Embedded minimal tori in $S^3$ and the Lawson conjecture, Acta Math. 211 (2013), no. 2, 177–190. MR 3143888, DOI 10.1007/s11511-013-0101-2
Simon Brendle, Minimal surfaces in $S^3$: a survey of recent results, Bull. Math. Sci. 3 (2013), no. 1, 133–171. MR 3061135, DOI 10.1007/s13373-013-0034-2
Simon Brendle, Two-point functions and their applications in geometry, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 581–596. MR 3237760, DOI 10.1090/S0273-0979-2014-01461-2
Simon Brendle and Otis Chodosh, On the maximum curvature of closed curves in negatively curved manifolds, Canad. Math. Bull. 58 (2015), no. 4, 713–722. MR 3415662, DOI 10.4153/CMB-2015-030-3
Kazuyuki Enomoto, A generalization of the isoperimetric inequality on $S^2$ and flat tori in $S^3$, Proc. Amer. Math. Soc. 120 (1994), no. 2, 553–558. MR 1163333, DOI 10.1090/S0002-9939-1994-1163333-7
Werner Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929), no. 1, 238–252 (German). MR 1512528, DOI 10.1007/BF01454836
Ailana Fraser and Martin Man-chun Li, Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary, J. Differential Geom. 96 (2014), no. 2, 183–200. MR 3178438
Michael E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225–1229. MR 726325, DOI 10.1215/S0012-7094-83-05052-4
Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127–133. MR 1656553, DOI 10.4310/AJM.1998.v2.n1.a2
Yu-Chu Lin and Dong-Ho Tsai, Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves, J. Evol. Equ. 12 (2012), no. 4, 833–854. MR 3000458, DOI 10.1007/s00028-012-0157-z
Shengliang Pan and Juanna Yang, On a non-local perimeter-preserving curve evolution problem for convex plane curves, Manuscripta Math. 127 (2008), no. 4, 469–484. MR 2457190, DOI 10.1007/s00229-008-0211-x
Joel L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501–506. MR 1163337, DOI 10.1090/S0002-9939-1994-1163337-4
Deyan Zhang, Inequalities with curvature and their stability estimates for convex curves, J. Math. Inequal. 10 (2016), no. 2, 433–443. MR 3455374, DOI 10.7153/jmi-10-34