1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

so

5. CENTER OF MASS AND NONLINEAR AVERAGES

We compute using the Jacobi equation

~ Jll = ~ (g (\le;erl, 1))

dr^2 dr Ill

g(\le;erl,1)

2

g(\le;er\le;erl,1) [\le;er1[

2

1113 + 111 + 111

9 (\l e;erl, 1)

2

g (R ( 1, 'Y) 'Y, 1) [\l e;er1[

2

1113 - 111 + 111 '

d2 (

dr2 Ill+ K l'Yl

2

Ill = lll-

1

lll

2

l'Yl

2

K - g (R (1, 'Y) 'Y, 1))

• lll-

3

(lll

2

[\l e;erl[

2

- 9 (\l e;erl, 1)

2

)

::::: 0,

177

where we used 1 (r) J_ 'Y (r) to conclude that lll^2 l'Yl^2 K -g (R (1, 'Y) 'Y, 1) 2::

0.

The corresponding ODE for¢ (r) is

¢" + K l'Yl^2 ¢ = 0

(l'YI = d(x,y) is a constant since"( is a geodesic) and has solutions

where

and

¢ (r) = ¢ (0) csKl'Yl2 (r) + ¢' (0) snKl'Yl2 (r),

{

~ sin (far)

sn"" (r) = r

~ sinh ( FK,r)

if K, > 0,

if K, = o,

if K, < 0,

{

cos (far) if"'> 0,

cs""(r)= 1 if,,,=0,

cosh ( FK,r) if"'< 0.

The functions sn"" (r) and cs"" (r) are the solutions to ¢" + "'</> = 0 with
sn"" (0) = 0, sn~ (0) = 1, cs"" (0) = 1, and cs~ (0) = 0.
Note that 1/ Ill is a unit vector (and has a limit as r --+ 0) and that

\le;erl(O) is well defined. From fr Ill= g (\le;erl, 1 ~ 1 ) we know that the

limit limr-+O+ fr Ill is also well defined. Now we compare Ill (r) with the

solution¢ (r) which satisfies¢ (0) = JlJ (0) = 0 and¢' (0) = limr-+O+ fr Ill.

Note that

¢ (r) = ¢' (0) snKl-Yl2 (r)