Centro-affine normal flows on curves: Harnack estimates and ancient solutions
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1189-1197.

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar p centro-affine normal flows are contracting origin-centered ellipses.

DOI : 10.1016/j.anihpc.2014.07.001
Classification : 53C44, 53A04, 52A10, 53A15
Mots-clés : Centro-affine normal flow, Affine differential geometry, Affine support function, Ancient solutions 
@article{AIHPC_2015__32_6_1189_0,
     author = {Ivaki, Mohammad N.},
     title = {Centro-affine normal flows on curves: {Harnack} estimates and ancient solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1189--1197},
     publisher = {Elsevier},
     volume = {32},
     number = {6},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.07.001},
     mrnumber = {3425259},
     zbl = {1329.53096},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.07.001/}
}
TY  - JOUR
AU  - Ivaki, Mohammad N.
TI  - Centro-affine normal flows on curves: Harnack estimates and ancient solutions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 1189
EP  - 1197
VL  - 32
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.07.001/
DO  - 10.1016/j.anihpc.2014.07.001
LA  - en
ID  - AIHPC_2015__32_6_1189_0
ER  - 
%0 Journal Article
%A Ivaki, Mohammad N.
%T Centro-affine normal flows on curves: Harnack estimates and ancient solutions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 1189-1197
%V 32
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.07.001/
%R 10.1016/j.anihpc.2014.07.001
%G en
%F AIHPC_2015__32_6_1189_0
Ivaki, Mohammad N. Centro-affine normal flows on curves: Harnack estimates and ancient solutions. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1189-1197. doi : 10.1016/j.anihpc.2014.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2014.07.001/

[1] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), 179 -197 | EuDML | MR | Zbl

[2] B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differ. Geom. 43 (1996), 207 -230 | MR | Zbl

[3] B. Andrews, Affine curve-lengthening flow, J. Reine Angew. Math. 506 (1999), 43 -83 | MR | Zbl

[4] B. Andrews, Motion of hypersurfaces by Gauss curvature, Pac. J. Math. 195 no. 1 (2000), 1 -34 | MR | Zbl

[5] S. Chen, Classifying convex compact ancient solutions to the affine curve shortening flow, J. Geom. Anal. (2013), http://dx.doi.org/10.1007/s12220-013-9456-z | MR

[6] C.M. Petty, Affine isoperimetric problems, Discrete Geometry and Convexity Ann. N.Y. Acad. Sci. 440 (1985), 113 -127 , http://dx.doi.org/10.1111/j.1749-6632.1985.tb14545.x | MR | Zbl

[7] M.N. Ivaki, Centro-affine curvature flows on centrally symmetric convex curves, Trans. Am. Math. Soc. (2014), arXiv:1205.6456v2 | MR | Zbl

[8] M.N. Ivaki, A. Stancu, Volume preserving centro-affine normal flows, Commun. Anal. Geom. 21 (2013), 671 -685 | MR | Zbl

[9] J. Loftin, M.P. Tsui, Ancient solutions of the affine normal flow, J. Differ. Geom. 78 (2008), 113 -162 | MR | Zbl

[10] E. Lutwak, The Brunn–Minkowski–Fiery theory II: affine and geominimal surface areas, Adv. Math. 118 (1996), 244 -294 | MR | Zbl

[11] G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79 -120 | MR | Zbl

[12] A. Stancu, Centro-affine invariants for smooth convex bodies, Int. Math. Res. Not. 2012 no. 10 (2012), 2289 -2320 | MR | Zbl

Cité par Sources :