Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Global analysis of the generalised Helfrich flow of closed curves immersed in $ \mathbb{R}^n$

Author: Glen Wheeler
Journal: Trans. Amer. Math. Soc. 367 (2015), 2263-2300
MSC (2010): Primary 53C44, 58J35
Published electronically: December 5, 2014
MathSciNet review: 3301865
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the evolution of regular closed elastic curves $ \gamma $ immersed in $ \mathbb{R}^n$. Equipping the ambient Euclidean space with a vector field $ \mathbbm {c}:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a function $ f:\mathbb{R}^n\rightarrow \mathbb{R}$, we assume the energy of $ \gamma $ is smallest when the curvature $ \vec {\kappa }$ of $ \gamma $ is parallel to $ \vec {c}_0 = (\mathbbm {c}\circ \gamma ) + (f\circ \gamma )\tau $, where $ \tau $ is the unit vector field spanning the tangent bundle of $ \gamma $. This leads us to consider a generalisation of the Helfrich functional $ \mathcal {H}^{\vec {c}_0}_{\lambda }$, defined as the sum of the integral of $ \vert\vec {\kappa }-\vec {c}_0\vert^2$ and $ \lambda $-weighted length. We primarily consider the case where $ f:\mathbb{R}^n\rightarrow \mathbb{R}$ is uniformly bounded in $ C^\infty (\mathbb{R}^n)$ and $ \mathbbm {c}:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is an affine transformation. Our first theorem is that the steepest descent $ L^2$-gradient flow of $ \mathcal {H}^{\vec {c}_0}_{\lambda }$ with smooth initial data exists for all time and subconverges to a smooth solution of the Euler-Lagrange equation for a limiting functional $ \mathcal {H}^{\vec {c}_\infty }_{\lambda }$. We additionally perform some asymptotic analysis. In the broad class of gradient flows for which we obtain global existence and subconvergence, there exist many examples for which full convergence of the flow does not hold. This may manifest in its simplest form as solutions translating or spiralling off to infinity. We prove that if either $ \mathbbm {c}$ and $ f$ are constant, the derivative of $ \mathbbm {c}$ is invertible and non-vanishing, or $ (f,\gamma _0)$ satisfy a `properness' condition, then one obtains full convergence of the flow and uniqueness of the limit. This last result strengthens a well-known theorem of Kuwert, Schätzle and Dziuk on the elastic flow of closed curves in $ \mathbb{R}^n$ where $ f$ is constant and $ \mathbbm {c}$ vanishes.

References [Enhancements On Off] (What's this?)

  • [1] Ben Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 20 (1997), 1001-1031. MR 1486693 (99a:58041),
  • [2] Ben Andrews, The affine curve-lengthening flow, J. Reine Angew. Math. 506 (1999), 43-83. MR 1665677 (2000e:53081),
  • [3] C. Baker,
    The mean curvature flow of submanifolds of high codimension,
    PhD thesis, Australian National University, 2011.
  • [4] Gerhard Dziuk, Ernst Kuwert, and Reiner Schätzle, Evolution of elastic curves in $ \mathbb{R}^n$: existence and computation, SIAM J. Math. Anal. 33 (2002), no. 5, 1228-1245 (electronic). MR 1897710 (2003f:53117),
  • [5] Charles M. Elliott and Harald Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl. 7 (1997), no. 1, 467-490. MR 1454678 (98h:58044)
  • [6] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. 28 (1973), no. 11, 693-703.
  • [7] Norihito Koiso, On the motion of a curve towards elastica, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 403-436 (English, with English and French summaries). MR 1427766 (98a:58037)
  • [8] T. Lamm,
    Biharmonischer wärmefluß, 2001.
    Diplomarbeit, Universität Freiburg.
  • [9] Joel Langer and David A. Singer, Curve-straightening in Riemannian manifolds, Ann. Global Anal. Geom. 5 (1987), no. 2, 133-150. MR 944778 (89i:58025),
  • [10] Joel Langer and David A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology 24 (1985), no. 1, 75-88. MR 790677 (86j:58023),
  • [11] Tsoy-Wo Ma, Higher chain formula proved by combinatorics, Electron. J. Combin. 16 (2009), no. 1, Note 21, 7. MR 2515761 (2010e:05017)
  • [12] Carlo Mantegazza and Luca Martinazzi, A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 857-874. MR 3060703
  • [13] A. Polden,
    Curves and surfaces of least total curvature and fourth-order flows,
    PhD thesis, Mathematisches Institut Universität Tübingen, 1996.
  • [14] Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525-571. MR 727703 (85b:58121),
  • [15] Yingzhong Wen, $ L^2$ flow of curve straightening in the plane, Duke Math. J. 70 (1993), no. 3, 683-698. MR 1224103 (94f:58028),
  • [16] Yingzhong Wen, Curve straightening flow deforms closed plane curves with nonzero rotation number to circles, J. Differential Equations 120 (1995), no. 1, 89-107. MR 1339670 (96e:58033),
  • [17] Glen Wheeler, On the curve diffusion flow of closed plane curves, Ann. Mat. Pura Appl. (4) 192 (2013), no. 5, 931-950. MR 3105957,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 58J35

Retrieve articles in all journals with MSC (2010): 53C44, 58J35

Additional Information

Glen Wheeler
Affiliation: Otto-von-Guericke-Universität, Postfach 4120, D-39016 Magdeburg, Germany
Address at time of publication: University of Wollongong, Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, Faculty of Engineering and Information Sciences, Northfields Avenue, Wollongong 2522, New South Wales, Australia

Received by editor(s): June 22, 2012
Published electronically: December 5, 2014
Additional Notes: Financial support from the Alexander-von-Humboldt Stiftung is gratefully acknowledged
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society