Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

Cheng, Da Rong; Zhou, Xin

doi:10.1090/tran/8510

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere
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by Da Rong Cheng and Xin Zhou PDF

Trans. Amer. Math. Soc. 374 (2021), 9007-9028 Request permission

Abstract:

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

References

Alberto Abbondandolo, Luca Asselle, Gabriele Benedetti, Marco Mazzucchelli, and Iskander A. Taimanov, The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere, Adv. Nonlinear Stud. 17 (2017), no. 1, 17–30. MR 3604943, DOI 10.1515/ans-2016-6003
Vladimir I. Arnold, Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow, 2004. Translated and revised edition of the 2000 Russian original; With a preface by V. Philippov, A. Yakivchik and M. Peters. MR 2078115
V. I. Arnol′d, The first steps of symplectic topology, Uspekhi Mat. Nauk 41 (1986), no. 6(252), 3–18, 229 (Russian). MR 890489
Luca Asselle and Gabriele Benedetti, The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles, J. Topol. Anal. 8 (2016), no. 3, 545–570. MR 3509572, DOI 10.1142/S1793525316500205
Luca Asselle and Felix Schmäschke, On the existence of closed magnetic geodesics via symplectic reduction, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 49, 28. MR 3766631, DOI 10.1007/s11784-018-0528-3
Abbas Bahri and Iskander A. Taimanov, Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2697–2717. MR 1458315, DOI 10.1090/S0002-9947-98-02108-4
Victor Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math. 4 (1993), no. 1, 1–10. MR 1209957, DOI 10.1142/S0129167X93000029
George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. MR 1501070, DOI 10.1090/S0002-9947-1917-1501070-3
Da Rong Cheng and Xin Zhou, Existence of constant mean curvature 2-spheres in riemannian 3-spheres, arXiv:2011.04136 (2020).
Tobias H. Colding and William P. Minicozzi II, Width and mean curvature flow, Geom. Topol. 12 (2008), no. 5, 2517–2535. MR 2460870, DOI 10.2140/gt.2008.12.2517
John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
Viktor L. Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 131–148. MR 1432462
Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. MR 979601, DOI 10.2307/1971486
Daniel Ketover and Yevgeny Liokumovich, On the existence of closed ${C}^{1,1}$ curves of constant curvature, arXiv:1810.09308, 2018.
L. Lyusternik and L. Šnirel′man, Topological methods in variational problems and their application to the differential geometry of surfaces, Uspehi Matem. Nauk (N.S.) 2 (1947), no. 1(17), 166–217 (Russian). MR 0029532
Fernando C. Marques and André Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017), no. 2, 577–616. MR 3674223, DOI 10.1007/s00222-017-0716-6
S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248 (Russian). MR 676612
S. P. Novikov and I. A. Taĭmanov, Periodic extremals of multivalued or not everywhere positive functionals, Dokl. Akad. Nauk SSSR 274 (1984), no. 1, 26–28 (Russian). MR 730159
Richard S. Palais, Critical point theory and the minimax principle, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 185–212. MR 0264712
Harold Rosenberg and Matthias Schneider, Embedded constant-curvature curves on convex surfaces, Pacific J. Math. 253 (2011), no. 1, 213–218. MR 2869442, DOI 10.2140/pjm.2011.253.213
Harold Rosenberg and Graham Smith, Degree theory of immersed hypersurfaces, Mem. Amer. Math. Soc. 265 (2020), no. 1290, v+62. MR 4080916, DOI 10.1090/memo/1290
Matthias Schneider, Closed magnetic geodesics on $S^2$, J. Differential Geom. 87 (2011), no. 2, 343–388. MR 2788659
Matthias Schneider, Alexandrov-embedded closed magnetic geodesics on $S^2$, Ergodic Theory Dynam. Systems 32 (2012), no. 4, 1471–1480. MR 2955323, DOI 10.1017/S0143385711000289
Michael Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), no. 1-2, 19–64. MR 926524, DOI 10.1007/BF02392272
Michael Struwe, Variational methods, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34, Springer-Verlag, Berlin, 2008. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 2431434
I. A. Taĭmanov, Non-self-intersecting closed extremals of multivalued or not-everywhere-positive functionals, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 2, 367–383 (Russian); English transl., Math. USSR-Izv. 38 (1992), no. 2, 359–374. MR 1133303, DOI 10.1070/IM1992v038n02ABEH002203
Xin Zhou and Jonathan Zhu, Existence of hypersurfaces with prescribed mean curvature I—generic min-max, Camb. J. Math. 8 (2020), no. 2, 311–362. MR 4091027, DOI 10.4310/CJM.2020.v8.n2.a2
Xin Zhou and Jonathan J. Zhu, Min-max theory for constant mean curvature hypersurfaces, Invent. Math. 218 (2019), no. 2, 441–490. MR 4011704, DOI 10.1007/s00222-019-00886-1
Xin Zhou and Jonathan J. Zhu, Min-max theory for networks of constant geodesic curvature, Adv. Math. 361 (2020), 106941, 16. MR 4043011, DOI 10.1016/j.aim.2019.106941