An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs W entropy

In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs WW entropy, the critical point of our entropy is the gradient shrinking soliton; however, there is no conjugate heat equation involved. This shows a close relation between Hamiltonʼs entropy and Perelmanʼs W entropy on closed surfaces.