1547671870-The_Ricci_Flow__Chow

70 3. SHORT TIME EXISTENCE

The general formula for the evolution of volume is given by the following
observation.

LEMMA 3.9. The volume element dμ evolves by

atdμ a = 2 l(··a lJ at9ij ) dμ = 2 H dμ.

PROOF. If {xi} ~=l are oriented local coordinates, then

dμ = Jdet9 dx^1 /\ · · · /\ dxn.

Hence

• a dμ = -l(a - log det g ) y ~ det g dx^1 /\ · · · /\ dx n.
  at 2 at

COROLLARY 3.10. The total scalar curvature f Mn Rdμ evolves by

:t (!Mn Rdμ) = !Mn (~RH -(Rc,h)) dμ.

We will also need to understand the evolution of length.

D

LEMMA 3.11. Let 'Yt be a time-dependent family of curves with fixed

endpoints in (Mn, g (t)), and let Lt ht) denote the length of "ft with respect

to the metric g (t). If

a

at^9 = h,

then

(3.8) : Lt("ft) = ~ 1 h (S, S) ds -1 (\! sS, V) ds,

t '"It '"It

where S is the unit tangent to "ft, V is the variation vector field, and ds is

the element of arc length.

PROOF. Consider the variation

'Y: [a,b] x (-c,c)--> Mn,

regarded as the family of curves 'Yt ( u) = 'Y ( u, t). Assume 'Y (a, ·) ::::::: x E Mn

and 'Y (b, ·)::::::: y E Mn. Define the vector fields

u =%(a/au) and v = 'Y* (a/at)

along 'Y, and set

s(u) ~ 1u (U,U)^112 du.

Then the unit tangent to 'Yt is

S ~ (U, U)-^1!^2 U.