- RELATION TO THE HARMONIC MAP FLOW 89
REMARK 3.26. Ben Andrews [4] has observed that when the universal
cover of the initial Riemannian manifold may be isometrically embedded as
a hypersurface in Euclidean or Minkowski 4-space, the Gauss curvature fl.ow
of that hypersurface effects the cross curvature fl.ow of the induced metric;
in this case, he can prove convergence results.
4.3. The DeTurck diffeomorphisms and the harmonic map flow.
Recall that the diffeomorphisms <pt defined by (3.29) and (3.35) satisfy
at a lfJt = -w o lfJ = gpq ( - r k pq + r -k) pq.
If :f is the Levi-Civita connection of a metric g, then it follows from Lemma
3.18 and equation (3.36) that
9 pq ( -rpq k + rpq -k ) -- ( ( lfJt -1) * g -) pq ( - r pq k + r -pq k ) -- ,6.g(t),!J lfJt
Therefore
a
at lfJt = ,6.g(t),g lfJt.Thus we come to the interesting observation that the DeTurck diffeomor-
phisms satisfy the harmonic map heat fl.ow:LEMMA 3.27. The diffeomorphisms 1.fJt used to obtain the solution g (t) of
the Ricci flow {3.32} from the solution g (t) of the Ricci- De Turck flow {3.33}satisfy the harmonic map flow with domain metric g ( t) and codomain metric
g.4.4. An alternate approach to existence and uniqueness. In
[63], Hamilton introduced a variant of DeTurck's argument for short-time
existence and uniqueness of the Ricci fl.ow on a closed manifold Mn. This
method uses both the Ricci- DeTurck fl.ow and the harmonic map fl.ow, and
provides a particularly elegant proof of uniqueness. Hamilton's strategy is
as follows:
STEP 1. Fix a background metric g on Mn.
STEP 2. If a solution to the Ricci fl.ow (3.32) exists, denote it by g (t).
In this case, let <pt denote the solution of the harmonic map heat fl.ow