Inverse curvature flow of closed Legendre curves

Abstract

In this paper, we deal with an inverse curvature flow of $\ell$-convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as $t \to \infty$. Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.