主要研究在曲线收缩流中的两个单调性公式.在几何流的研究中,熵的单调性起着重要的作用.Gage和Hamilton首先研究了在平面曲线收缩流中的曲率玻尔兹曼熵的一阶和二阶导数,并定义了一个除非在收缩圆上否则严格递增的单调性公式,称之为W熵,它是以弧长s为参数的.另一个则是Hamilton 在研究曲线收缩流的紧致的古典解分类时提出的一个单调性公式,称其为 I 公式.以角度 θ为参数对这两个单调公式进行证明.
A Note on Two Monotonicity Formulas in Curve Shortening Flow
This paper mainly studies two monotonicity formulas in curve shortening flow.In the study of geometric flow,the monotonicity of entropy plays an important role.Gage and Hamilton first studied the first and second derivatives of the curvature Boltzmann entropy in the plane curve shortening flow,and defined a monotonicity formula that is strictly increasing unless on the contraction circle.It is called W entropy and takes the arc length s as a parameter.The other is a monotonicity formula put forward by Hamilton when studying the classification of compact classical solutions of curve shortening flow,which is called I formula.In this paper,these two monotonicity formulas with angle θ as a parameter are proved.
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