1547671870-The_Ricci_Flow__Chow

78 3. SHORT TIME EXISTENCE

proves that K<;, = At, hence that dim K<;, = n (n - 1) /2. Now we further

consider the kernel of 0-[D (Reg)]((). For all h EK<;,, it follows easily from
(3.10) that

hence that

1

0-[D (Reg)](()= -2 1 (1^2 idK<.

In particular, 0-[D (Reg)] (()l.K< : K<;, --t K<;, is an automorphism in each

fiber where ( i= 0, which proves that

. n(n- 1)

dim(im(O-[D(Rcg)](()))~d1mK<;,=

2

.

By (3.23), this implies that

(3.26) dim (ker (0-[D (Reg)](()))= n.

Before we conclude this discussion, it is worth comparing the global and

infinitesimal viewpoints it affords. The facts that

im 8;.l ker 8g

under the global inner product (3.20), and that

C^00 (S2T* Mn)= im8; EB ker8g

are classical. (See [ 38 ] and [ 15 ].) On the other hand, define

I<;,~ im (0-[8;J (()) = ker (0-[D (Reg)](()).

Then in each fiber where ( i= 0, we have

S2T* Mn = I<;,+ K<;,.

However, this decomposition is not orthogonal with respect to the inner

product (·, ·) induced by g on each fiber. This is because the operator

0-[D (Reg)](() is not self-adjoint: indeed, if h, k E S2T* Mn, we have

(0-[D (Reg)](() (h), k) - (h, 0-[D (Reg)](() (k))

1

=

2

((tr gk) h - (tr gh) k, (@ ().

3. The Ricci-DeTurck flow and its parabolicity

The short exact sequence (3.25) show that the nonlinear differential op-

erator Reg is not an elliptic operator on the metric g. Because of this, we

cannot immediately apply standard theory to conclude that a unique solu-

tion of the Ricci flow exists for a short time. In spite of this fact, the Ricci.

flow does enjoy short-time existence and uniqueness:

THEOREM 3. 13 (Hamilton). If (Mn,go) is a closed Riemannian mani-

fold, there exists a unique solution g (t) to the Ricci flow defined on some

positive time interval [O, c) such that g (0) = go.