Area-preserving evolution of nonsimple symmetric plane curves


The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles, while for Abresch–Langer type curves, the convergence to m-fold circles happens if and only if the enclosed algebraic area is positive.

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Correspondence to Xiao-Liu Wang.

Additional information

This work is partially supported by the National Natural Science Foundation of China 11101078, 11171064 and the Education Department Program of Liaoning Province L2010068.

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Wang, XL., Kong, LH. Area-preserving evolution of nonsimple symmetric plane curves. J. Evol. Equ. 14, 387–401 (2014).

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Mathematics Subject Classification (2010)

  • Primary 53C44
  • 35B40
  • Secondary 35K59
  • 37B25


  • Mean curvature flow
  • nonlocal parabolic equation
  • asymptotic behavior