1547671870-The_Ricci_Flow__Chow

5. THE RICCI FLOW AS A HEAT EQUATION 91

Existence of harmonic coordinates is a straightforward consequence of
standard existence theory for elliptic partial differential equations.

THEOREM 3.29 (Local solvability of elliptic PDE). Let

F(x, u, ou, ... , oku)

be a C^00 function, and let Xo E !Rn. If there exists a function v E ck ( U)

defined in a neighborhood U of xo such that

F(x, v, ov, ... , okv )(xo) = 0

and if F is elliptic at v {that is, if the linearization of F at v is elliptic),

then there exists a neighborhood V of xo and a function u such that

F(x, u, ou,. .. , oku)(x) = 0

for all x E V and

aJu(xo) = [)Jv(xo)

for all multi-indices j such that U I < k.

COROLLARY 3.30. Given a point p E M, there exist harmonic coordi-

nates defined in some neighborhood of p.

PROOF. Let {xi} denote geodesic coordinates centered at p. Because

I'fj(P) = 0, we have

~xi(p) = [gij (8i8j - rtak)xi] (p) = o.

By the theorem above, there exist a neighborhood V of p and functions { ui}

such that ~ui = 0 holds in V, with ui(p) = 0 and Ojui(p) = Ojxi(p) = 8].

As a consequence, { ui} are harmonic coordinates in some neighborhood

W c V of p. D

Standard elliptic regularity theory implies that if the metric g is in a

Holder class Ck,a (respectively cw) in the original coordinates, then the

harmonic coordinates themselves are in the class ck+l,a (respectively cw).

In particular, the map from the original coordinates to the harmonic co-

ordinates is of class Ck+l,a (Cw). Since transforming a tensor involves at

most first derivatives of the map, it follows readily that g itself has optimal

regularity in harmonic coordinates.

LEMMA 3.31. If a metric g is of class Ck,a (Cw) in a given coordinate

chart, then any tensor of class ck,a (cw) in that chart belongs to the same

Holder class in harmonic coordinates. In particular, g itself is of class Ck,a

(Cw).

The insight into the Ricci fl.ow which harmonic coordinates provide stems

from the following observation.