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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

 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.
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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