1547671870-The_Ricci_Flow__Chow

4. RELATION TO THE HARMONIC MAP FLOW 85

Hence

\7 (df) = L (\7df)0 dxi 0 dxj 0

0

° a

i,J,a .. y

where

The harmonic map Laplacian with respect to the domain metric g
and codomain metric h is defined to be the trace

namely

.. a

~ g, hf= (gi^3 \7·d·f'Y) i J [)yr -.

In components,

Given Jo : Mn ---+ Nm, the harmonic map fl.ow introduced by Eells
and Sampson is

(3.40a)

(3.40b)

of_~

at - g,h^1 '

f (0) = fo.

This is a parabolic equation, so a unique solution exists for a short time.
When f is a diffeomorphism, the Laplacian ~g,hf may be rewritten in
a useful way:

LEMMA 3.18. Let f: (Mn,g)---+ (Nn,h) be a diffeomorphism of Rie-

mannian manifolds. Then

PROOF. We first compute the Christoffel symbols of the pull-back of a

metric in local coordinates {xi} on Mn and {ya} on Nn. Let <.p : Mn ---+ Nn