On Gage-Hamilton's entropy formula and Harnack inequality for the curve shortening flow

By the first two derivatives of the Boltzmann entropy of the curvature, which was first studied by Gage and Hamilton for the curve shortening flow in the plane, we define a monotonicity formula which is strictly increasing unless on a shrinking circle. By calculating pointwisely we give an alternate proof of Gage-Hamilton's Harnack inequality.