Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his "Curvature Difference Flow", 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.

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For a class of general quasilinear equations on S 1 , we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution.

We study the contraction of a convex immersed plane curve with speed 1 α k α , where α ∈ (0, 1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α = 1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]). * AMS Subject Classifications: 35K15, 35K55. We thank Ben Andrews for giving us this summary.rescaled about the final point (the isoperimetric ratio approaches infinity); and the exceptional ones where the isoperimetric ratio remains bounded converge to homothetic solutions, which have been classified. For α > 1/3, the rescaled solutions converge to circles; and for α = 1/3, they converge to ellipses.Remark 2 As a consequence of Theorem 1, we have the following interesting elliptic result. For 0 < λ < 3 (here λ = 1/α), the only positive 2π-periodic solution to the equation