Uma demonstração do teorema de Grayson sobre evolução de curvas planas pela curvatura

Abstract

Based on the recent work by Andrews and Bryan [2] we present a new proof of the celebrated Grayson's theorem [4], which describes the asymptotic behavior of simple curves evolving by the curve shortening flow. The proof represents a remarkable simplification over the previous methods and consist of normalizing the flow in order to preserve the length (equal to 2 ). It is then established an isoperimetric inequality which provides a lower bound for the length of chords in terms of the corresponding arcs and elapsed time. This estimate is suficiently strong to uniformly control the curvature in time, implying, without many difficulties, that the curvature of the normalized flow converges in the C∞ topology to the function identically equal to 1.