1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

30 1. RICCI SOLITONS

PROOF. In these coordinates, the sectional curvatures are

1-(w')^2 Y^2 - (n -.l)X^2 (X^2 + Y^2 + n>.W^2 - fo,X)

Zll -- w 2 - - n(n - l)W (^2) ' v2 = - nW2 '
for 2-planes tangent to and perpendicular to, respectively, the orbits of the
rotational symmetry. The evolution equations for the numerators are

-^1 -(Y d 2 ( - n - 1 )X^2 ) = (X^2 - >. W^2 - aX ) ( 2 ( Y - n - 1 ) X 2)
2 ds

• a(n - l)X(-(X^2 + Y^2 + n>.W^2 - vnX))

and

! .!!:._ ( -( X^2 + Y^2 + n>. W^2 - Jn X))

2 ds

= -(X^2 ~ >.W^2 + ~(aX - l))(X^2 + Y^2 + n>.W^2 - vnX)

+ ~X(Y^2 - (n - l)X^2 ).

2

Observing that the sign of the first nonconstant factor in the second line

of each equation is positive, we conclude that the signs v1 > 0, v2 > 0 and

v1 < 0, v2 < 0 are both preserved. D

We will now obtain the asymptotic behavior of the curvature relative to

distance r.

PROPOSITION 1.42. Assuming that the sectional curvatures are negative,

then they both decay at least quadratically in r, and for large r, the warping

function w(r) is bounded between two linear functions of r.

PROOF. First, we establish that, as the trajectories approach the origin,
the distance r is asymptotic to a constant times 1/W. We will do this by
establishing a positive lower bound for W^2 / X, by examining the equations

.!!:._ (w2) = w2 (x2->.w2+1-(aY2 + >...;nw2))

ds X X X '

.!!:._ (y2) = y2 (x2 - >.w2+1-2aX - (aY2 + >...;nw2))

dsX X X.

For r sufficiently large, we can bound the nonconstant terms in the paren-
theses (those that do not involve dividing by X) by a small number c:. Then,
if we set Y ~ aY^2 /X and W ~ >.fo,W^2 /X, we have

! (W + Y) 2: (W + Y) (1 -E - y - w).

Hence, W + Y 2: c for some positive constant c. On the other hand, because

sectional curvature v1 is negative, Y^2 / X:::; (n-l)X, and so lims-t+oo Y = 0.