Lifespan theorem and gap lemma for the globally constrained Willmore flow

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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

Abstract
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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 and 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.
[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.
[3]	E. Kuwert and R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom., 57 (2001), 409-441.
[4]	E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.
[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.
[6]	J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.
[7]	J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().
[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.
[9]	G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.
[10]	G. Wheeler, "Fourth Order Geometric Evolution Equations," Ph.D thesis, University of Wollongong, 2010. doi: 10.1017/s0004972710001863.
[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.
[12]	T. Willmore, "Riemannian Geometry," Oxford University Press, New York, 1993. doi: 10.2307/3612154.

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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.
[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.
[3]	E. Kuwert and R. Schätzle, The Willmore flow with small initial energy, J. Differential Geom., 57 (2001), 409-441.
[4]	E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.
[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.
[6]	J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.
[7]	J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().
[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.
[9]	G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.
[10]	G. Wheeler, "Fourth Order Geometric Evolution Equations," Ph.D thesis, University of Wollongong, 2010. doi: 10.1017/s0004972710001863.
[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.
[12]	T. Willmore, "Riemannian Geometry," Oxford University Press, New York, 1993. doi: 10.2307/3612154.

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