232 6. ENTROPY AND NO LOCAL COLLAPSINGSince Wc:(g, f, T) = JM (T (R + .6.f) - c(J - n)) (47rT)-nl^2 e-f dμ, inte-
grating this by parts and applying the divergence theorem, we havedds Wc:(g, f, T)
-Vij (Rij + "\li"\ljf) e-f
+ (R + .6.f) e-f (¥ -h - n22 ~)
~Lr M(e-f) (h-.Jf)
+~(-Eh-c (f-n) (¥ -h-~;)) e-f-TVij (~j + "\Ji"\Jjf + ; 7 9ij) e-f
+T ( R + 2.6.f - 1'7 fl^2 ) e-f ( ¥ -h - ~;)
-E (j - n - 1) e-f (¥ -h-~n
+( (R + .6.f) e-f + (~ ( .6.f - 1'7 fl2 + ~) e-f
Now, integrating by parts tells us again that the terms on the last line areJM ( ((R+ .6.f) + ~ ( .6.f-1'7fl
2
+ ~)) e-f(47rT)-%dμ= JM ( ( R + .6.f + ~~) udμ.
Substituting this into the above formula yields (6.9).In the next two exercises we give a unified proof of the monotonicity
formulas for entropy, energy, and expander entropy.
EXERCISE 6.14 (Monotonicity of We: from the first variation formula).
When we require that the variation ( v, h, () satisfies ( = c and ¥-h-~ ( =
0 in (6.30) (e.g., preserves the measure (47rT)-nl^2 e-f dμ), this leads to the
following gradient fl.ow:
(6.32)(6.33)(6.34)f)
fJt9ij = -2 (Rij + "\li"\ljf),
8f nc
at = -.6.f - R-2T'
dT