The Combination Curve Flow On The Plane

This paper mainly investigates two kinds of convex curve flow on the plane whichconsist of area-preserving curve flow and length-preserving curve flow. First,we considerthe following evolving problem: Let X(λ, t):[a, b]×(0,+∞)â†'R2be a family of closedplane curves with X(λ,0)=X0(λ) being a closed,strictly convex curve.We will prove this flow decreases the length of the curve but increases the enclosed areaduring the evolution process. As time t goes to infinity, the limiting curve will be a finitecircle(i.e.,a circle with finite radius) in the C∞metric.Next we discuss the second curve flow. Suppose Y (μ, t):[c, d]×(0,+∞)â†'R2is afamily of closed curves on the plane and Y (μ,0)=Y0(μ) is a closed,strictly convex planecurve. Now we consider the following geometric evolving problem:The above curve flow keeps its length a constant and expands the area. It exists globallyand as the time goes to infinity it converges to a circle in the C∞metric.