- NOTES AND COMMENTARY 219
By limt->oo <lJf (t) = 0 and (5.71), we have
dE ()Q d^2 E f 00 dE
dt (0) = - lo dt2 (t) dt:::::; 2K lo dt (t) dt
= 2Klog ( Vol~M) JM udμ) JM udμ-2KE(O).
Hence- JM u IV log ul
2
dμ :::::; 2K log (Vol ~M) JM udμ) JM udμ- 2K JM ulogudμ
and the proposition follows. D- Notes and commentary
Subsection 1.1. As we remarked earlier, the function 1 is also known as
the dilaton; in the physics literature there are numerous references to Perel-
man's energy functional (see Green, Schwarz, and Witten [162], Polchinksi
[307], Strominger and Vafa [341] for example), although Perelman is the
first to consider it in the context of Ricci flow. The Ricci flow is the 1-loop
approximation of the renormalization group fl.ow (see Friedan [145]).
Subsection 1.2. For a computational motivation for fixing the mea-
sure, see also §4 in Chapter 2 of [111], where Perelman's functional is moti-
vated starting from the total scalar curvature functional. In particular, let
8 g = v. The variation of the total scalar curvature is
8 JMRdμ= JM (div(divv)-~V-Rc·v+R~)dμ
= - JM (Re - ~ g) · vdμ.
This says that V (JM Rdμ) = - Re +~g, where the gradient is calculated
with respect to the standard L^2 -metric. To try to find a functional :F with
V :F = -Re, we want to get rid of the ~ g term. Now this term is due to
the variation of dμ. So we consider the distorted volume form e-f dμ and
assume its variation is 0. Hence
8 JM Re-f dμ =JM (8R) e-f dμ =JM (div (divv) - ~V - Re ·v) e-f dμ
and now we have the extra terms JM (div (divv) - ~V) e-f dμ. We compen-
sate for this by considering
8 JM IV 112
e-f dμ =JM (8IV112
) e-f dμ=JM (-v (V 1, V 1) + V f · VV) e-f dμ,