1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

12 1. RICCI SOLITONS

PROOF. Let ( M^2 , g) be a Riemannian surface (not necessarily complete)
with a Killing vector field K. Let x E M and define a smooth unit speed

path 'Y: (ro - c:, r 0 + c:) ---+ M by 'Y (r) = l~~~j 1 ('Y (r)), 'Y (ro) = x. Define a

1-parameter family of smooth paths f3r : (Bo - c:, Bo+ c:) ---+ M by /Jr (B) =

K (f3r (B)), f3r (Bo) = 'Y (r), and a 1-parameter family of smooth unit speed

paths 'Ye : (ro - c:, ro + c:)---+ M by 'Ye (r) = l~~~j 1 ('Ye (r)), 'Ye (ro) = f3ro (B):

Note that 'Yeo = 'Y·
CLAIM.
f3r (e) ='Ye (r).
PROOF OF CLAIM. This follows from

[

J (K) ] 1 1 1 2

K, IJ (K)I = IJ (K)I [K, J (K)] - 211 (K)l3K IJ (K)I J (K) = 0

since [K, J (K)] = 0 and K IJ (K)l^2 = 0.

Hence (r, e) defines local coordinates on a neighborhood of x. Define

f (r) ~ IKI (f3r), where we are using the fact that IKI is constant on f3r·

The metric is given by

I J ( K) J ( K) ) 2 I J ( K) ) 2

g = \ IJ (K)I' IJ (K)I dr + 2 \ K, IJ (K)I drde + (K, K) de

= dr^2 + f (r)^2 dB^2 •

D

It can be shown on a surface that the integral curves of a Killing vector

field J (V' f) are closed loops. In particular one can prove the following

without using the Uniformization Theorem (see [86]).

THEOREM 1.19 (Shrinking surface soliton is spherical). If (M^2 , g) is a

shrinking gradient soliton on a closed surface, then g has constant positive

curvature.

More generally, if (M^2 , g) is a shrinking gradient soliton on a closed,

orientable 2-dimensional orbifold with isolated singularities, then g is rota-
tionally symmetric with positive curvature. When the 2-orbifold is bad, i.e.,
not covered by a smooth surface, a unique Ricci soliton exists, which is a
nontrivial shrinking gradient Ricci soliton (see [180] and [372]). Topologi-
cally, a closed, bad 2-orbifold has either one or two singular points [343].

REMARK 1.20. There do not exist complete shrinking Ricci solitons on

noncompact surfaces.

3.2. Warped products. Many known examples of gradient Ricci soli-
tons have been constructed in the form of warped products. In particular,
we consider a metric on the product Mn+l =Ix Nn of the form
(1.37) g = dr^2 + w(r)2g,