1547671870-The_Ricci_Flow__Chow

92 3. SHORT TIME EXISTENCE

LEMMA 3.32. In harmonic coordinates, the Ricci tensor is given by

• 2Rj = ~ (gij) + Qij (g-^1 , og),
  where ~ (9ij) denotes the Laplacian of the component 9ij of the metric re-
  garded as a scalar function, and Q denotes a sum of terms which are qua-
  dratic in the metric inverse g-^1 and its first derivatives og.
  PROOF. The Ricci tensor is given in local coordinates by

-2Rjk = -2R~jk = - 2 ( oqr]k - ajr~k + r;kqp - r~kr]p)

= -Oq [gqr (Oj9kr + Ok9jr - Or9jk)]

+ oj [gqr (8q9kr + ok9qr - Or9qk)] + r;kqP - r~krJp·

Using Lemma 3.1, we write this as

• 2Rj k = gqr (oqor9jk - oqok9jr + OjOk9qr - OjOr9qk) + g-^1 g-^1 og og
  = ~ (9jk) - gqrak (r~r9sj) - gqraj (r~r9sk) + g -^1 g-^1 og og,
  where g-^1 g-^1 og * og denotes a sum of contractions whose exact formula
  is irrelevant. In harmonic coordinates, the second and third terms of the
  last line vanish, and we obtain


• 2Rjk = ~ (9jk) + g-^1 g-^1 og * a 9.
  D

COROLLARY 3.33. In harmonic coordinates, the Ricci flow takes the form

of a system of nonlinear heat equations for the components of the metric

tensor:

(3.43) at9ij a = ~ (9ij) + Qij ( g -1 , og ).

REMARK 3.34. The reader is cautioned that equation (3.43) is not tenso-

rial. Since the metric is parallel, it is of course clear that ~g = 0. Moreover,

one should not in general expect coordinates which are harmonic at time

t = 0 to be harmonic at times t > 0.

Notes and commentary

Hamilton's original proof of short-time existence [58] used the Nash-

Moser implicit function theorem. [57] DeTurck's proof [36, 37] is a sub-

stantial simplification. Other presentations of DeTurck's proof may be found

in §6 of [63] and in [112].

The Lichnerowicz Laplacian acting on symmetric 2-tensors (which as
we saw above is essentially the linearization of the Ricci fl.ow operator) is
formally the same as the Hodge- de Rham Laplacian acting on 2-forms. See
Remark A.3 in Appendix A.