1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

44 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(3) f'v (so) = expx (V), and

( 4) gr / r=O Irv ( S) = V ( s) for V E B ( c).

For example, we may define

!'v (s) = exp,,(s) (v (s))

for s E [O, so] and VE B (c-).

We have

Lg bv);:::: dg (expx (V) ,xo) for VE B (c-),

Lg (1' 0 ) = dg (x, xo).

This implies that the C^00 function f : B (x, c-) -+ lR+ defined by

is an upper barrier for dg ( ·, xo), that is,

f(y);::::dg(y,xo) foryEBg(x,c-),

f (x) = dg (x, xo).

Thus, in the barrier sense of Definition 18.5, we have

(18.6)

Now we turn to bounding !::ig f ( x) from above. Let { E1, ... , En-I, f'^1 (so)}

be an orthonormal basis of TxM, with respect to g. Then

{ E1 ( s) , ... , En-I ( s) , /'^1 ( s) }

is an orthonormal basis of T,,(s)M for s E [O, so]. Recall that the second

variation in the direction V of the length functional Lg, which holds for

families of continuous piecewise smooth paths, is given by

oi Lg(!') ~ ~22 I Lg (Irv)

ur r=O

= foso ( / V' I' V ( s) / 2 - ( R ( f'^1 ( s), V ( s)) V ( s), !'^1 ( s))) ds

(18.7) = foso (((' (s))

2

/V (s)i2-(^2 (R (!''(s), V (s)) V (s) ,f'^1 (s))) ds

for any V E B (c-) (regarding the vanishing of the 'endpoint' term in the

second variation formula, recall that Irv (so) = expx ( r V) is a constant

speed geodesic passing through x); in the above formula, R ( · , · ) · denotes

the Riemann curvature (3, 1)-tensor.