278 6. ENTROPY AND NO LOCAL COLLAPSING
If g is a steady gradient soliton flowing along 'V f, then
Rij + 'Vi'Vjf = 0,
2Af-J'VJJ^2 + R = 0,(6.109)Writing the second variation this way, we see the appearance of the linear
trace Harnack quadratic.
Note that the first line of (6.109) is independent of h whereas for (point-
wise) measure-preserving variations (v, h) of (g, f), the last two lines of
(6.109) vanish. This shows that given Vij, the variation (v, h) minimizing
J: 2 :F(g, J) while preserving the (integral) constraint JM e-f dμ = 1 has the
last two lines of (6.109) nonpositive; not surprisingly, the last two lines are
less than or equal to a negative norm squared, as we shall show below.
Using (6.108), the second line in (6.109) may be rewritten as
- 2 JM [ \Ji\Jj ( h - ~)] Vije-f dμ
= 2 JM (div v - v ('V f)) · [ \l ( h - ~)] e-f dμ.
Hence for a steady gradient soliton g flowing along 'V f,
::2:F(g, f) =JM ( \liVjp - ~\JpVij) (\lpVij) e-f dμ
- 2 JM (div v - v ('VJ)) · [ \l (h -~)] e-f dμ
+2JMl\l(h-~)l
2
e-fdμ.Completing the square, we have
::2 :f'(g, j) = JM ( \JiVjp - ~ \J pVij) (\J pVij) e-f dμ
-~ r Jdivv-v('Vf)J^2 e-fdμ
2}M
- 2 JM I~ (divv -v ('VJ))+ \l ( h-~) j
2
e-f dμ.