A Combination Curvature Flow

In this thesis, we will follow the ideas of Gage [7], [8], Gage-Hamilton [9] and Pan [14] to deal with a new curvature flow for convex plane curves, this flow is a "convex combination" of the area-preserved flow of Gage[8] and the flow defined by Pan[14]. Let X(u, t) = (x(u,t),y(u,t)) : [a,b] x [0,∞)→ R² be a family of planar simple closed curves, X(u, 0) = X₀(u) also a planar simple closed curve. The equations for the flow are the following:where the subscript denotes the derivative about t, λ is a positive constant.We will prove that this flow shortens the length of the evolving curves but expands the area bounded by the curves and will make the evolving curves more and more circular during the evolving process. The final shape of the curve will be a circle.The thesis consists of three chapters. In the first one, we will briefly talk about the history and background of curvature flow theory and some basic facts about convex planar curves. In chapter two, we will give the evolution quations geometric quantities for general planar curve flows. The third part is the main part of the present thesis in which we will first give a new model of evolution for convex plane curves and then show that convex curves remain so during the evolution process and the final shape is circular in the Hausdorff metric. We will prove that the evolution problem is equivalent to an initial value problem of a certain nonlinear differential-integral quantion system and then use the maximum principle and the Laray-Schauder fixed point theorem to prove existence and uniqueness of the claasi-cal solutions in the local sense. Then the proofs of global existence and uniqueness for the claasical solutions to the initial value problem.The last two sections are devoted to proving the "C²" and "C^∞" convergence of the evolving curves to a circle.