Geodesic
Closed ideal planar curves
Andrews, Ben1 ; McCoy, James2 ; Wheeler, Glen3 ; Wheeler, Valentina-Mira3
1 Applied and Nonlinear Analysis Group, Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia
2 Priority Research Centre for Computer-Assisted Research Mathematics and its Applications, School of Mathematical & Physical Sciences, University of Newcastle, Callaghan, NSW, Australia
3 Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, Australia
Geometry & topology, Tome 24 (2020) no. 2, pp. 1019-1049.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the L2 sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply covered circle. Moreover, we show that curves in any homotopy class with initially small L3∥ks∥22 enjoy a uniform length bound under the flow, yielding the convergence result in these cases.

DOI : 10.2140/gt.2020.24.1019
Classification : 35K25, 53C44, 58J35
Keywords: constant mean curvature, curvature flow, geometric evolution equation

Andrews, Ben 1 ; McCoy, James 2 ; Wheeler, Glen 3 ; Wheeler, Valentina-Mira 3

1 Applied and Nonlinear Analysis Group, Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia
2 Priority Research Centre for Computer-Assisted Research Mathematics and its Applications, School of Mathematical & Physical Sciences, University of Newcastle, Callaghan, NSW, Australia
3 Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, Australia 
@article{GT_2020_24_2_a6,
     author = {Andrews, Ben and McCoy, James and Wheeler, Glen and Wheeler, Valentina-Mira},
     title = {Closed ideal planar curves},
     journal = {Geometry & topology},
     pages = {1019--1049},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2020},
     doi = {10.2140/gt.2020.24.1019},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1019/}
}
TY  - JOUR
AU  - Andrews, Ben
AU  - McCoy, James
AU  - Wheeler, Glen
AU  - Wheeler, Valentina-Mira
TI  - Closed ideal planar curves
JO  - Geometry & topology
PY  - 2020
SP  - 1019
EP  - 1049
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1019/
DO  - 10.2140/gt.2020.24.1019
ID  - GT_2020_24_2_a6
ER  - 
%0 Journal Article
%A Andrews, Ben
%A McCoy, James
%A Wheeler, Glen
%A Wheeler, Valentina-Mira
%T Closed ideal planar curves
%J Geometry & topology
%D 2020
%P 1019-1049
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1019/
%R 10.2140/gt.2020.24.1019
%F GT_2020_24_2_a6
Andrews, Ben; McCoy, James; Wheeler, Glen; Wheeler, Valentina-Mira. Closed ideal planar curves. Geometry & topology, Tome 24 (2020) no. 2, pp. 1019-1049. doi : 10.2140/gt.2020.24.1019. http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1019/

[1] C Baker, The mean curvature flow of submanifolds of high codimension, PhD thesis, Australian National University (2010)

[2] R Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 (2003) 572 | DOI

[3] A Dall’Acqua, P Pozzi, A Spener, The Łojasiewicz–Simon gradient inequality for open elastic curves, J. Differential Equations 261 (2016) 2168 | DOI

[4] G Dziuk, E Kuwert, R Schätzle, Evolution of elastic curves in Rn : existence and computation, SIAM J. Math. Anal. 33 (2002) 1228 | DOI

[5] G Harary, A Tal, 3D Euler spirals for 3D curve completion, from: "Computational geometry", ACM (2010) 393 | DOI

[6] G Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984) 237 | DOI

[7] G Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math. 382 (1987) 35 | DOI

[8] J Mccoy, The surface area preserving mean curvature flow, Asian J. Math. 7 (2003) 7 | DOI

[9] J Mccoy, G Wheeler, Y Wu, A sixth order curvature flow of plane curves with boundary conditions, from: "2017 MATRIX annals" (editors D R Wood, J de Gier, C E Praeger, T Tao), MATRIX Book Ser. 2, Springer (2019) 213 | DOI

[10] S Parkins, G Wheeler, The polyharmonic heat flow of closed plane curves, J. Math. Anal. Appl. 439 (2016) 608 | DOI

[11] G Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations 44 (2012) 131 | DOI

[12] G Wheeler, On the curve diffusion flow of closed plane curves, Ann. Mat. Pura Appl. 192 (2013) 931 | DOI

Cité par Sources :