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Tytuł artykułu

A convergence result for the Gradient Flow of ∫ |A|2in Riemannian Manifolds

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EN
Abstrakty
EN
We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.
Twórcy
  • Institut für Angewandte Mathematik, Westfälische–Wilhelms Universität Münster,
    Einsteinstrasse 62, 48149 Münster, Germany
Bibliografia
  • [1] P. Breuning. Immersions with bounded second fundamental form. J. Geom. Anal., doi 10.1007/s12220-014-9472-7[Crossref][WoS]
  • [2] R. Bryant. A duality theorem for Willmore surfaces. J. Diff. Geom., 20:23–53, 1984
  • [3] A. A. Cooper. A compactness theorem for the second fundamental form. arXiv:1006.5697
  • [4] A. Huber. On subharmonic functions and differential geometry in the large. Comm. Math. Helv., 32:13–72, 1957.[Crossref]
  • [5] E. Kuwert and R. Schätzle. The Willmore flow with small initial energy. J. Diff. Geom., 57:409–441, 2001.
  • [6] E. Kuwert and R. Schätzle. Gradient flow for the Willmore functional. Commun. Anal. Geom., 10(2):307–339, 2002.
  • [7] E. Kuwert and R. Schätzle. Removability of point singularities of Willmore surfaces. Ann. ofMath. (2), 160(1):315–357, 2004.
  • [8] E. Kuwert and R. Schätzle. Closed Surfaces with bounds on their Willmore energy. Annali Sc. Norm. Sup. Pisa. Cl. Sci.11:605–634, 2012.
  • [9] E. Kuwert, A. Mondino and J. Schygulla. Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds. Math. Ann., 359(1):379–425.[WoS]
  • [10] F. Link Gradient flow for the Willmore functional in Riemannian manifolds with bounded geometry. PhD Thesis, Albert-Ludwigs-Universität Freiburg, arXiv:1308.6055.
  • [11] J. Metzger, G. Wheeler and V.M. Wheeler. Willmore flow of surfaces in Riemannian spaces I: Concentration-compactness.arXiv:1308.6024
  • [12] A. Mondino. Existence of Integral m−Varifolds minimizing |A|p and |H|p in Riemannian Manifolds. Calc. Var., 49(1-2):431–470, 2014.[Crossref]
  • [13] A. Mondino. Some results about the existence of critical points for theWillmore functional. Math. Z., 266(3):583–622, 2010.[WoS]
  • [14] A. Mondino. The conformal Willmore Functional: a perturbative approach. J. Geom. Anal., 23(2):764-811, 2013.[WoS][Crossref]
  • [15] A. Mondino and T. Riviére. Immersed spheres of finite total curvature into manifolds. Adv. Calc. Var., 7(4):493–538, 2014.[WoS]
  • [16] A. Mondino and T. Riviére. Willmore Spheres in Compact Riemannian Manifolds. Adv. Math., 232(1):608–676, 2013.[WoS]
  • [17] S. Müller and V. Šverák. On surfaces of finite total curvature. J. Diff. Geom., 42:229–258, 1995.
  • [18] L. Simon. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom., 1(2):281–325, 1993.
  • [19] G. Simonett. The Willmore flow near spheres. Differential Integral Equations, 14(8):897–1024, 2001.
  • [20] B. White. Complete surfaces of finite total curvature. J. Diff. Geom., 26:315–326, 1987.
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Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_1515_geofl-2015-0001
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