1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

468 A. BASIC RICCI FLOW THEORY

3.4. Surface entropy formulas. The surface entropy N is defined for
a metric of strictly positive curvature on a closed surface M^2 by

(Vl-p. 133) N(g) ~ f RlogRdμ.

}M2

Let f be the potential function, defined up to an additive constant by

(Vl-5.8) l:!.f = R-r.

(See Lemma 5.38 on p. 133 and Proposition 5.39 on p. 134 of Volume One.)

PROPOSITION A.45 (Surface entropy formula). If (M^2 , g (t)) is a solu-

tion of the normalized Ricci flow on a compact surface with R (·, 0) > 0,

then

(Vl-5.25)

(Vl-p. 134)

3.5. Ancient 2-dimensional solutions.
3.5.1. Examples. (See pp. 24-28 in Section 2 of Chapter 2 in Volume
One.)
Hamilton's cigar soliton is the complete Riemannian surface (IR^2 , gI',),
where

(Vl-2.4)

. dx ® dx + dy ® dy
gI', -;- 1 + x2 + y2
This manifold is also known in the physics literature as Witten's black
hole. In polar coordinates

(Vl-2.5)

dr^2 + r^2 d()^2

gI', =

1 +r^2

If we define

(Vl-p. 25a) s ~ arcsinh r =log (r + Vl + r^2 ),

then we may rewrite gI', as

(Vl-2. 7) gI', = ds^2 + tanh^2 s d()^2.

The scalar curvature of gI', is

4 4 16

RI', = --= - ------=-

1 + r^2 cosh^2 s (e^8 + e-^8 )^2 ·

(Vl-p. 25b)

(See pp. 31-34 in subsection 3.3 of Chapter 2 in Volume One.) Let h
be the flat metric on the manifold M^2 = IRxSt, where st is the circle of
radius 1. Give M^2 coordinates x E IR and () E st = IR/2nZ. The Rosenau