1547671870-The_Ricci_Flow__Chow

88 3. SHORT TIME EXISTENCE

EXERCISE 3.23. Verify that

( C -1)ij \JiCjk = 2 1 ( C -1)ij \JkCij·

Use this to prove that if the sectional curvatures of (M^3 , g) are negative

everywhere (or positive everywhere), then c 9 is a metric and

is a harmonic map.

Since we are interested in curvature flows, one may consider the cross-

curvature fl.ow. The cross curvature flow, is defined on a 3-manifold of

negative sectional curvature by

and on a 3-manifold of positive sectional curvature by

When M^3 is closed, it can be proven that a solution exists for a short

time. The main open problem is to determine the long-time existence and

convergence properties of the cross curvature fl.ow. When the initial metric

has negative sectional curvature, the following monotonicity formulas have

been proven by Hamilton and the first author.

PROPOSITION 3.24. As long as a solution to the cross curvature flow

exists, one has

8

• at Vol(E) > - 0

and

• d^1 [1 - (lJEi1).. - (det --E) dμ :S 0.

1

/

3

]

dt M3 3 detg

Note that Vol (E) is scale-invariant in g. The integrand

~ (gij Eij) - ( det E / det g) l/^3

is the difference between the arithmetic and geometric means of - K 1 , - K 2 ,

and - K3. It vanishes identically if and only if the sectional curvature is

constant. The monotonicity of the integral is surprising, because the metric

g is expanding, and the integral scales like g^112.

CONJECTURE 3.25. If (M^3 ,go) is a manifold of negative sectional cur-

vature, the cross curvature flow with initial condition go should exist for all

positive time and converge.