- SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS 259
in (23.36). Using (23.127) and (23.128), we then have near t = O that^13
JM rJ (log HN +~log (47rt)) Hd~ + 0 (t3/2)=JM rJ (-d
2
~' y) + log¢o + ~~t + 0 (t^2 )) Hdμ + 0 (t^312 )
=JM rJ (-d
2
~' y) + 11
2 Rpq (y) xPxq +^0 (r (x)(^3) )) Hdμ
+JM rJ (~R (y) + 0 (r (x))) tHdμ + 0 (t^312 )
= -~ + ~tR (y) + 0 (t^312 ).
In particular, by (23.130),(
a A) (l H n 1 ( 4 )) logHN+~log(47rt)I
at + ux og N + 2 og 7rt + t
x=y- (t + :t) JM rJ (logHN +~log (47rt)) Hdμ
=--+-R(y)--n^1 1 ( --+-tR(y) n^1 ) --a ( -tR(y)^1 ) +O ( t^1 ;^2 )
2t 2 t 2 3 at 3= -~R (y) + 0 (t^1!^2 ).
REMARK 23.35. In contrast, we shall see in §4 of the next chapter that,
by a result of Perelman, a corresponding quantity for the Ricci fl.ow is o (1).
PROBLEM 23.36. Calculate the asymptotics near t = 0 of - JM f Hdμ.
5. Supplementary material: Elementary tools
5.1. Expansion for the determinant of a square matrix.
Here we recall an elementary result used in the derivation of the first
four terms of (23.114).
LEMMA 23.37 (Expansion for the determinant). If a square matrix has
the expansion(23.138)(^13) Note that lim.,_,o x log x = 0.