Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence of good sweepouts on closed manifolds

Authors: Longzhi Lin and Lu Wang
Journal: Proc. Amer. Math. Soc. 138 (2010), 4081-4088
MSC (2010): Primary 53C22; Secondary 58J35
Published electronically: May 26, 2010
MathSciNet review: 2679629
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we establish estimates for the harmonic map heat flow from $ S^1$ into a closed manifold, and we use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.

References [Enhancements On Off] (What's this?)

  • 1. G.D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199-300. MR 1501070
  • 2. G.D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, RI, 1927. MR 0209095
  • 3. T.H. Colding and C. De Lellis, The min-max construction of minimal surfaces, in Surv. Differ. Geom. VIII, Intl. Press, Somerville, MA, 2003. MR 2039986 (2005a:53008)
  • 4. T.H. Colding and W.P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain $ 3$-manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005), no. 3, 561-569. MR 2138137 (2006c:53068)
  • 5. T.H. Colding and W.P. Minicozzi II, Width and mean curvature flow, Geom. Topol. 12 (2008), no. 5, 2517-2535. MR 2460870 (2009k:53165)
  • 6. T.H. Colding and W.P. Minicozzi II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537-2586. MR 2460871 (2009k:53166)
  • 7. C.B. Croke, Area and the length of the shortest closed geodesic, J. Differential Geom. 27 (1988), no. 1, 1-21. MR 0918453 (89a:53050)
  • 8. M.A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71-111. MR 0979601 (90a:53050)
  • 9. M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants, Math. Z. 185 (1984), no. 1, 23-43. MR 0724044 (86b:35089)
  • 10. R.S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Math., vol. 471, Springer-Verlag, Berlin-New York, 1975. MR 0482822
  • 11. F. Hélein, Harmonic maps, conservation laws and moving frames, Cambridge Tracts in Math., vol. 150, Cambridge University Press, Cambridge, 2002. MR 1913803 (2003g:58024)
  • 12. G.M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996 (revised edition 2005). MR 1465184 (98k:35003)
  • 13. L. Lin, Closed geodesics in Alexandrov spaces of curvature bounded from above, to appear in J. Geom. Anal.
  • 14. S.K. Ottarsson, Closed geodesics on Riemannian manifolds via the heat flow, J. Geom. Phys. 2 (1985), no. 1, 49-72. MR 0834094 (87g:58024)
  • 15. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three manifolds, arXiv:math.DG/0307245.
  • 16. M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), no. 4, 558-581. MR 0826871 (87e:58056)
  • 17. M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer-Verlag, Berlin, 1996. MR 1411681 (98f:49002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C22, 58J35

Retrieve articles in all journals with MSC (2010): 53C22, 58J35

Additional Information

Longzhi Lin
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218

Lu Wang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): October 8, 2009
Received by editor(s) in revised form: February 4, 2010
Published electronically: May 26, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society