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Transactions of the American Mathematical Society

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Centro-affine curvature flows on centrally symmetric convex curves


Author: Mohammad N. Ivaki
Journal: Trans. Amer. Math. Soc. 366 (2014), 5671-5692
MSC (2010): Primary 53C44, 53A04, 52A10, 53A15; Secondary 35K55
DOI: https://doi.org/10.1090/S0002-9947-2014-05928-X
Published electronically: July 21, 2014
MathSciNet review: 3256179
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Abstract: We consider two types of $ p$-centro-affine flows on smooth, centrally symmetric, closed convex planar curves: $ p$-contracting and $ p$-expanding. Here $ p$ is an arbitrary real number greater than $ 1$. We show that, under any $ p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $ \pi $ converge, in the Hausdorff metric, to the unit circle modulo $ SL(2)$. As a $ p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $ p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized to enclose constant area $ \pi $, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to $ SL(2)$ transformations, to the unit circle. At the end of the paper, we present a new proof of the $ p$-affine isoperimetric inequality, $ p\geq 1$, for smooth, centrally symmetric convex bodies in $ \mathbb{R}^2$.


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  • [1] Ben Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151-171. MR 1385524 (97b:53012), https://doi.org/10.1007/BF01191340
  • [2] Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207-230. MR 1424425 (97m:58045)
  • [3] Ben Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315-371. MR 1660843 (99k:58038), https://doi.org/10.1007/s005260050111
  • [4] Ben Andrews, The affine curve-lengthening flow, J. Reine Angew. Math. 506 (1999), 43-83. MR 1665677 (2000e:53081), https://doi.org/10.1515/crll.1999.008
  • [5] Ben Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1-34. MR 1781612 (2001i:53108), https://doi.org/10.2140/pjm.2000.195.1
  • [6] Sigurd Angenent, Guillermo Sapiro, and Allen Tannenbaum, On the affine heat equation for non-convex curves, J. Amer. Math. Soc. 11 (1998), no. 3, 601-634. MR 1491538 (99d:58039), https://doi.org/10.1090/S0894-0347-98-00262-8
  • [7] Andrea Cianchi, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 419-436. MR 2551138 (2010h:46041), https://doi.org/10.1007/s00526-009-0235-4
  • [8] S. Ivanov, Private communication.
  • [9] Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187-204. MR 0030135 (10,719b)
  • [10] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759 (88d:35005)
  • [11] John Loftin and Mao-Pei Tsui, Ancient solutions of the affine normal flow, J. Differential Geom. 78 (2008), no. 1, 113-162. MR 2406266 (2009e:53084)
  • [12] Monika Ludwig and Matthias Reitzner, A classification of $ {\rm SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219-1267. MR 2680490 (2011g:52025), https://doi.org/10.4007/annals.2010.172.1223
  • [13] Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244-294. MR 1378681 (97f:52014), https://doi.org/10.1006/aima.1996.0022
  • [14] Erwin Lutwak and Vladimir Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), no. 1, 227-246. MR 1316557 (95m:52007)
  • [15] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Optimal Sobolev norms and the $ L^p$ Minkowski problem, Int. Math. Res. Not. , posted on (2006), Art. ID 62987, 21. MR 2211138 (2007d:52007), https://doi.org/10.1155/IMRN/2006/62987
  • [16] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Cramér-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information, IEEE Trans. Inform. Theory 51 (2005), no. 2, 473-478. MR 2236062 (2008a:94056), https://doi.org/10.1109/TIT.2004.840871
  • [17] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $ L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38. MR 1987375 (2004d:46039)
  • [18] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), no. 1, 59-81. MR 1890647 (2003f:52006), https://doi.org/10.1215/S0012-9074-02-11212-5
  • [19] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $ L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132. MR 1863023 (2002h:52011)
  • [20] Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248 (96e:53014)
  • [21] C. M. Petty, Affine isoperimetric problems, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 113-127. MR 809198 (87a:52014), https://doi.org/10.1111/j.1749-6632.1985.tb14545.x
  • [22] L. A. Santaló, An affine invariant for convex bodies of $ n$-dimensional space, Portugaliae Math. 8 (1949), 155-161 (Spanish). MR 0039293 (12,526f)
  • [23] Guillermo Sapiro and Allen Tannenbaum, Image smoothing based on an affine invariant flow, in: Proceedings of Conference on Information Sciences and Systems, Johns Hopkins University, March, 1993
  • [24] Guillermo Sapiro and Allen Tannenbaum, Affine invariant scale-space, Internat. J. Comput. Vision 11, 25-44, 1993.
  • [25] Guillermo Sapiro and Allen Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), no. 1, 79-120. MR 1255274 (94m:58049), https://doi.org/10.1006/jfan.1994.1004
  • [26] Guillermo Sapiro and Allen Tannenbaum, On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42 (1993), no. 3, 985-1009. MR 1254129 (94m:58048), https://doi.org/10.1512/iumj.1993.42.42046
  • [27] Rolf Schneider, Convex bodies: The Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • [28] Carsten Schütt and Elisabeth Werner, The convex floating body, Math. Scand. 66 (1990), no. 2, 275-290. MR 1075144 (91i:52005)
  • [29] Alina Stancu, The floating body problem, Bull. London Math. Soc. 38 (2006), no. 5, 839-846. MR 2268369 (2007k:52011), https://doi.org/10.1112/S0024609306018728
  • [30] Alina Stancu, Two volume product inequalities and their applications, Canad. Math. Bull. 52 (2009), no. 3, 464-472. MR 2547812 (2011b:52011), https://doi.org/10.4153/CMB-2009-049-0
  • [31] Alina Stancu, Centro-affine invariants for smooth convex bodies, Int. Math. Res. Not. IMRN 10 (2012), 2289-2320. MR 2923167, https://doi.org/10.1093/imrn/rnr110
  • [32] Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399-422. MR 1757001 (2001h:53016), https://doi.org/10.1007/s002220000059
  • [33] Neil S. Trudinger and Xu-Jia Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of Math. (2) 167 (2008), no. 3, 993-1028. MR 2415390 (2010h:35168), https://doi.org/10.4007/annals.2008.167.993
  • [34] Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882. MR 812353 (87e:53009), https://doi.org/10.1002/cpa.3160380615

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Additional Information

Mohammad N. Ivaki
Affiliation: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada H3G 1M8
Email: mivaki@mathstat.concordia.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-05928-X
Received by editor(s): September 17, 2011
Received by editor(s) in revised form: June 11, 2012
Published electronically: July 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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