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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 & 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.  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.  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.   Google Scholar [4] E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307.   Google Scholar [5] Y. N. Liu, Gradient flow for the Helfrich functional,, Chin. Ann. Math. B, 33 (2012), 931.  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.   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.  doi: 10.1007/s00209-010-0720-7.  Google Scholar [9] G. Simonett, The Willmore flow near spheres,, Differential Integral Equations, 14 (2001), 1005.   Google Scholar [10] G. Wheeler, "Fourth Order Geometric Evolution Equations,", Ph.D thesis, (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.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar [12] T. Willmore, "Riemannian Geometry,", Oxford University Press, (1993).  doi: 10.2307/3612154.  Google Scholar

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##### References:
 [1] G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35.  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.  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.   Google Scholar [4] E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307.   Google Scholar [5] Y. N. Liu, Gradient flow for the Helfrich functional,, Chin. Ann. Math. B, 33 (2012), 931.  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.   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.  doi: 10.1007/s00209-010-0720-7.  Google Scholar [9] G. Simonett, The Willmore flow near spheres,, Differential Integral Equations, 14 (2001), 1005.   Google Scholar [10] G. Wheeler, "Fourth Order Geometric Evolution Equations,", Ph.D thesis, (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.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar [12] T. Willmore, "Riemannian Geometry,", Oxford University Press, (1993).  doi: 10.2307/3612154.  Google Scholar
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