1547671870-The_Ricci_Flow__Chow

3. POSITIVE SECTIONAL CURVATURE DOMINATES 259

u

FIGURE l. Boundary of the region Kin the (u, v)-plane.

Now consider the map <P: (!\^2 T* M^3 @s /\^2 T* M^3 ) ___, ffi.^2 defined by

<P : IP' f-+ ( u (IP') , v (IP')) ~ (lvl , tr IP')

and observe that lP' E K if and only if <P (IP') E K. (The condition v ?: - 3u

is satisfied automatically, because tr lP' ?: 3v (IP').) Thus the proof will be
complete if we show that <P (slP' + (1 - s) Q) E K for all IP', Q E K and
s E [O, 1]. Since trace is a linear functional,

v (slP'+ (1-s)Q) = sv(lP') + (1-s)v (Q).

Because v ?: -3u and the function - u (IP') = min1w1= 1 lP' (W, W) is concave,

(9.3) -v (slP'+ ~ - s)Q) ~ u(slP'+ (1- s)Q) ~ su(lP') + (1- s)u(Q).

Together, these relations show that <P (slP' + (1 - s) Q) EH, where His the

rhombus with vertices (u(lP'),v(lP')), (u(Q),v(Q)), (-~v(lP'),v(lP')), and

(-iv(Q),v(Q)). Because each vertex of H lies in the convex set K, we

have H c K. 0

LEMMA 9 .6. Given any x E M^3 and quadratic form

lP' E ( /\ 2T* M3 ®s /\ 2T* M3) x

with eigenvalues >.(IP') ?: μ(IP') ?: v (IP'), where v (IP') < 0, define

tr lP'

n (IP')~ -v (IP') - log (- v (IP')).

Let M ( t) correspond to Rm [g]. Then wherever n (M) is defined,

d

dt n (M) ?: -v (M).