the main objective of this paper is to study the boundedness character, the periodicity character, the convergenceand the global stability of the positive solutions of the nonlinear rational difference equation/ , n 0,1,2,....0 01      kii n ikin i n i x  x b  xwhere the coefficients i i b , ,  together with the initial conditions ,.... , , 1 0 x x x k  are arbitrary...

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to existence (Giga's conjecture) the shape of Type I singularities (Chou's conjecture).