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March  2014, 13(2): 715-728. doi: 10.3934/cpaa.2014.13.715

Lifespan theorem and gap lemma for the globally constrained Willmore flow

Yannan Liu 1, and  Linfen Cao 2,

1. 

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China
2. 

College of Mathematics and Information Science, Henan Normal University, Henan, 453007

Received  March 2013 Revised  July 2013 Published  October 2013

We study a fourth-order flow, which can be seen as a globally constrained Willmore flow. We obtain a lower bound on the lifespan of the smooth solution, which depends on the concentration of curvature for the initial surface and the constrained term. We also give a gap lemma for this flow, which is an important lemma in the study of the blowup analysis.
Keywords: Geometric evolution equations, fourth order, Willmore flow..
Mathematics Subject Classification: Primary: 35J60, 35K45, 52K44,53A0.
Citation: Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
References:
[1]

G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.  Google Scholar

[2]

H. Y. Jian and Y. N. Liu, Long-time existence of mean curvature flow with external force fields, Pacific J. Math., 234 (2008), 311-324. doi: 10.2140/pjm.2008.234.311.  Google Scholar

[3]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom., 57 (2001), 409-441.  Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.  Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional, Chin. Ann. Math. B, 33 (2012), 931-940. doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  Google Scholar

[7]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().   Google Scholar

[8]

J.McCoy, G. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178. doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.  Google Scholar

[10]

G. Wheeler, "Fourth Order Geometric Evolution Equations," Ph.D thesis, University of Wollongong, 2010. doi: 10.1017/s0004972710001863.  Google Scholar

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698. doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

T. Willmore, "Riemannian Geometry," Oxford University Press, New York, 1993. doi: 10.2307/3612154.  Google Scholar

show all references

References:
[1]

G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.  Google Scholar

[2]

H. Y. Jian and Y. N. Liu, Long-time existence of mean curvature flow with external force fields, Pacific J. Math., 234 (2008), 311-324. doi: 10.2140/pjm.2008.234.311.  Google Scholar

[3]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom., 57 (2001), 409-441.  Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.  Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional, Chin. Ann. Math. B, 33 (2012), 931-940. doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  Google Scholar

[7]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().   Google Scholar

[8]

J.McCoy, G. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178. doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.  Google Scholar

[10]

G. Wheeler, "Fourth Order Geometric Evolution Equations," Ph.D thesis, University of Wollongong, 2010. doi: 10.1017/s0004972710001863.  Google Scholar

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698. doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

T. Willmore, "Riemannian Geometry," Oxford University Press, New York, 1993. doi: 10.2307/3612154.  Google Scholar

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