The linear trace Harnack quadratic on a steady gradient Ricci soliton   satisfies the heat equation
  We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton. 
Bennett Chow and Peng Lu
math.DG