1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

360 7. THE REDUCED DISTANCE

where the exponential map is restricted to inside the cut locus in the tangent

space. Then the volume form of g on M\ (Cut (p) U {p}) is given by

dμ9 = J bv (s)) drJ\ dO".

The volume forms of the geodesic spheres S (p, r) ~ { x EM : d (x,p) = r},

at smooth points, are given by

dO"s(p,r) = J bv (s)) dO".

The Jacobian is related to the mean curvatures of the geodesic spheres and

the Ricci curvatures of the metric g on M by the following formulas:

0

or logJ = H

and

_§_H =-Re(_§___, _§_)-1hl^2

or or or

< - Re (_§_ §) - H2

• or' or n-l'
  where h is the second fundamental form of S (p, r). The Bishop-Gromov
  volume comparison theorem may be proved this way (see [111] for example).
  Now we turn to the case of Ricci fl.ow. Let IV ( T), T E [O, T), be an
  £-geodesic emanating from p with limr__,o y'T~ ( T) = V. Let JY ( T) , i
  1, ... , n, be £-Jacobi fields along Iv with
  Jr (0) = 0 and ('VvJr) (0) = Ef,

where { Ef} : 1 is an orthonormal basis for TpM with respect tog (0). Note

that Jr ( T) is a smooth function of V and T > 0 since g ( T) is smooth.

Via the orthonormal basis { Ef} ~=l we can identify TpM with IR_n. Since

D (.Cexp(V, T)) (Ef) = Jr (T) (see Lemma 7.92), the Jacobian of the £-

exponential map .CJv (T) E IR. (called the £-Jacobian for short) is the

square root of the determinant (computed using the inner products on the

tangent spaces from the Riemannian metric g (T)) of the basis of £-Jacobi

fields:

(J[ (T), ... , J;: (T)).

That is,

£Jv(T)~ det(\Jr(T),Jl(T))) ·

g(r) nxn

It is clear that .CJv (T) is a smooth function of (V,T) when T > 0. Another
equivalent way of describing .CJ v ( T) is to define

.CJv(T)dx(V) ~ [(.Crexp(V))*dμg(r,L'. 7 exp(V))J,

where dx is the standard Euclidean volume form on (TpM,g(O,p)).