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DEFORMING A STARSHAPED CURVE INTO A CIRCLE BY AN AREA-PRESERVING FLOW

Part of: Parabolic equations and systems Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  23 April 2020

JIANBO FANG Open the ORCID record for JIANBO FANG [Opens in a new window]
JIANBO FANG*
Affiliation:
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, PR China email fjbwcj@126.com
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Abstract

We show that a class of area-preserving flows can deform every starshaped curve into a circle.

Keywords

area-preserving flowstarshaped curvedivergent quasi-linear parabolic equation

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35K05: Heat equation 53A04: Curves in Euclidean space
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 498 - 505
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The work is supported by the National Natural Science Foundation of China (Nos. 11861004 and 11561012) and the Introducing Talents Project in Guizhou University of Finance and Economics (2018YJ102).

References

Angenent, S. B., ‘Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature’, Ann. of Math. (2) 132 (1990), 451–483.CrossRefGoogle Scholar
Angenent, S. B., ‘Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions’, Ann. of Math. (2) 133 (1991), 171–215.10.2307/2944327CrossRefGoogle Scholar
Chao, X. L., Ling, X. R. and Wang, X. L., ‘On a planar area-preserving curvature flow’, Proc. Amer. Math. Soc. 141 (2013), 1783–1789.10.1090/S0002-9939-2012-11745-9CrossRefGoogle Scholar
Chou, K. S. and Zhu, X. P., The Curve Shortening Problem (CRC Press, Boca Raton, FL, 2001).CrossRefGoogle Scholar
Chow, B., Liou, L. P. and Tsai, D. H., ‘Expansion of embedded curves with turning angle greater than -𝜋’, Invent. Math. 3 (1996), 415–429.Google Scholar
Chow, B. and Tsai, D. H., ‘Geometric expansion of convex plane curves’, J. Differential Geom. 2 (1996), 312–330.CrossRefGoogle Scholar
Dittberner, F., ‘Curve flows with a global forcing term’, Preprint, 2018, arXiv:1809.08643.Google Scholar
Gage, M. E., ‘Curve shortening makes convex curves circular’, Invent. Math. 76 (1984), 357–364.CrossRefGoogle Scholar
Gage, M. E., ‘On an area-preserving evolution equation for plane curves’, in: Nonlinear Problems in Geometry, Contemporary Mathematics, 51 (ed. DeTurck, D. M.) (American Mathematical Society, Providence, RI, 1986), 51–62.CrossRefGoogle Scholar
Gage, M. E. and Hamilton, R. S., ‘The heat equation shrinking convex plane curves’, J. Differential Geom. 23 (1986), 69–96.CrossRefGoogle Scholar
Gao, L. Y. and Pan, S. L., ‘Star-shaped curves under the CSF and Gage’s area-preserving flow’, in preparation.Google Scholar
Grayson, M. A., ‘The heat equation shrinks embedded plane curves to round points’, J. Differential Geom. 26 (1987), 285–314.CrossRefGoogle Scholar
Guan, P. F. and Li, J. F., ‘A mean curvature type flow in space forms’, Int. Math. Res. Not. IMRN 13 (2015), 4716–4740.CrossRefGoogle Scholar
Huisken, G., ‘The volume preserving mean curvature flow’, J. reine angew. Math. 382 (1987), 35–48.Google Scholar
Mayer, U. F., ‘A singular example for the averaged mean curvature flow’, Exp. Math. 10 (2001), 103–107.CrossRefGoogle Scholar
Oaks, J. A., ‘Singularities and self-intersections of curves evolving on surfaces’, Indiana Univ. Math. J. 43 (1994), 959–981.10.1512/iumj.1994.43.43042CrossRefGoogle Scholar
Tsai, D. H., ‘Geometric expansion of starshaped plane curves’, Comm. Anal. Geom. 3 (1996), 459–480.CrossRefGoogle Scholar