On Area-preserving and Length-preserving Nonlocal Flow of Convex Closed   Plane Curves
  For any $\alpha>0,$ we study $k^{\alpha}$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}% $-norm.$\ $Other relevant $k^{\alpha}$-type nonlocal flow is also discussed when $\alpha\geq1.\ $ 
Dong-Ho Tsai, Xiao-Liu Wang
math.DG math.AP