Some Remarks on Ricci Flow and the Quantum Potential

We indicate some formulas connecting Ricci flow and Perelman entropy to Fisher information, differential entropy, and the quantum potential. 1. FORMULAS INVOLVING RICCI FLOW Certain aspects of Perelman’s work on the Poincaré conjecture have applications in physics and we want to suggest a few formulas in this direction; a fuller exposition will appear in a longer paper [11] and in a book in preparation [8]. We go first to [14, 25, 26, 27, 28, 29, 34, 40] and simply write down a few formulas from [29, 40] here with minimal explanation. Thus one has Perelman’s functional (1.1) F = ∫ M (R+ |∇f |)exp(−f)dV and a so-called Nash entropy (1A) N(u) = ∫ M ulog(u)dV where u = log(−f). One considers Ricci flows g = gt with δg ∼ ∂tg = h and for (1B) 2u = −∂tu−∆u+Ru = 0 (or equivalently ∂tf+∆f−|∇f |2+R = 0) it follows that ∫ M exp(−f)dV = 1 is preserved ∂tN = F. Note the Ricci flow equation is ∂tg = −2Ric and when g(t) = a(t)φ∗t (g(0)) for diffeomorphisms φt : M → M g(t) is called a Ricci soliton. If φt is generated by a vector field X, i.e. ∂tφt(p) = X ◦φt then g(t) = φ∗t (g(0)) is called a gradient steady soliton when Xi = ∇if . In such a situation one shows in [29] that (1.2) Ric+Hess(f) = 0; R+∆f = 0; |∇f | +R = const.; ∂tf = |∇f | Extremizing F via δF ∼ ∂tF = 0 involves Ric + Hess(f) = 0 or Rij + ∇i∇jf = 0 and one knows also that (1.3) ∂tN = ∫ M (|∇f | +R)exp(−f)dV = F;