1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. 1-AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW 31

LEMMA 17.28 (Lower bound for the maximum value of a minimizer).

On a closed manifold we have

(17.88) mtixwT 2: exp (2Rmin T (g) - n 4 log(47rT) - n 2 - 2μ^1 (g, T) ).

PROOF. By (17.15), a minimizer wT satisfies

T (-4~wT + RwT) - wT log (w;) - (~ log(47rT) + n) wT = μ (g, T) wT.

At a point xT EM where wT attains its maximum, we have (~wT) (xT) ~ 0,

so that

Hence

mfiXWT =WT (xT) 2: exp ( ~R-~ log(47rT) - ~ - ~μ (g, T))

and the lemma follows. D

4. 1-and 2-loop variation formulas related to RG flow

In this section we discuss formulas related to Perelman's energy func-
tional and its variation.^12 One may wish that some of these formulas are
related to 'renormalization group fl.ow' (RG fl.ow) in physics; the 'loop' ter-
minology is from there. However the point of view we take is simply to
calculate first variation formulas for certain Riemannian geometric invari-
ants and to look for structure in these formulas. We leave the calculations
as exercises for the reader.

4.1. Some 1-loop formulas.

Let (Mn, g) be a closed Riemannian manifold and let f be a function

on M. Recall from (17.1) that Perelman's energy functional is

(17.89)

(we add the subscript 1 to F with the hope that this is the first in an infinite
sequence of functionals). The integrand in (17.89) appears in a contracted
second Bianchi-type identity (see §1.3 of [152]):

(17.90) div ((Re +\7\7 f) e-f) = ~e-f\7 ( R + 2~f - l\7 !1^2 ).

Let v be a symmetric 2-tensor on Mand let X be a vector field on M.

The linear trace Harnack quadratic is given by

(17.91) L (v, X) ~div (divv) + \v, Re) - 2 (divv, X) + v (X, X)

(^12) We would like to thank Shengli Kong for helpful discussions.