Remarks on some isoperimetric properties of thek− 1 flow

Abstract

We consider the evolution of a convex closed plane curve γ 0 along its inward normal direction with speed k − 1, where k is the curvature. This flow has the feature that it is the gradient flow of the (length − area) functional and has been previously studied by Chou and Zhu, and Yagisita. We revisit the flow and point out some interesting isoperimetric properties not discussed before.We first prove that if the curve γ t converges to the unit circle S 1 (without rescaling), its length L(t) and area A(t) must satisfy certain monotonicity properties and inequalities.On the other hand, if the curve γ t (assume γ 0 is not a circle) expands to infinity as t → ∞ and we interpret Yagisita's result in the right way, the isoperimetric difference L 2 (t) − 4π A(t) of γ t will decrease to a positive constant as t → ∞. Hence, without rescaling, the expanding curve γ t will not become circular. It is asymptotically close to some expanding curve C t , where C 0 is not a circle and each C t is parallel to C 0 . The asymptotic speed of C t is given by the constant 1.