等仿射曲线收缩流的Harnack不等式

在仿射空间中研究了基于等仿射曲线收缩流的一族闭凸等仿射曲线的Harnack不等式.首先，根据仿射空间中等仿射曲线的几何演化性质定义一类新的闭凸等仿射曲线Harnack量，进而得到该Harnack量满足的几何演化方程.其次，利用最大值原理证明Harnack量为非负，即给出闭凸等仿射曲线的Harnack不等式，并得到Harnack量中参数的相应约束条件.然后，利用新定义的Harnack量进一步研究了闭凸等仿射曲线的Hamilton’s Harnack不等式.最后基于闭凸等仿射曲线Harnack不等式和柯西-施瓦兹(Cauchy-Schwarz)不等式推导出了经典的Harnack不等式.

Harnack Inequality for Equi-Affine Curve Shortening Flow

The Harnack inequalities of a family of closed convex equi-affine curves based on equi-affine curve shortening flow were studied. Firstly， according to the geometric evolution property of the equi-affine curve， a new type of Harnack quantity on the closed convex equi-affine curve was defined， then the evolution equation about the Harnack quantity of the closed convex equi-affine curve was discovered. Secondly， by the maximum principle， the non-negativity of the Harnack quantity， i.e.， the Harnack inequality of the closed convex equi-affine curve， was investigated. Moreover， the constraint conditions of parameters in the Harnack quantity were found. Then， the Hamilton’s Harnack inequality of the closed convex equi-affine curve was further explored using the newly defined Harnack quantity.Finally， the classical Harnack inequality was derived based on Harnack inequality of the closed convex equi-affine curve and Cauchy-Schwarz inequality.