- PARABOLICITY OF THE RICCI-DETURCK FLOW 79
REMARK 3.14. We shall see in Corollary 7.7 that the lifetime of a max-
imal solution is bounded below by
c
maxMn IRm [go]l 90 '
where c is a universal constant depending only on n.
Hamilton's original proof [58] of this result relied on the Nash- Moser in-
verse function theorem and was lengthy and technically difficult. However,
he showed that the degeneracy of the equation is due only to its diffeo-
morphism invariance. Soon after, DeTurck found a much simpler proof of
Theorem 3.13. DeTurck showed that it is possible to modify the Ricci flow
and thereby obtain a parabolic PDE by a clever trick: one modifies the right-
hand side of the equation by adding a term which is a Lie derivative of the
metric with respect to a certain vector field which in turn depends on the
metric. Remarkably, one then can obtain a solution to the original Ricci flow
equation by pulling back the solution of the modified flow by appropriately
chosen diffeomorphisms.
To motivate how the DeTurck trick is done, we rewrite the linearization
of the Ricci tensor as follows:
(3.27)
- 2 [D (Re 9 ) (h)]jk = ~ hjk
- \J j (gPq\J qhpk) - \J k(gPq\J qhpj) + \J j \J k(gpq hqp)
- 2gqp R;jk hrp - gqp Rjp hkq - gqp Rkp hjq·
Define a 1-form V = V (g, h) by V = B 9 (h) where B 9 is given by (3.24), so
that
Vik-'-gPQ\J h k - - \Jk(gpqh^1 )
...,... pq 2 pq·
We may then express the linearization of the Ricci tensor as
(3.28)
where the symmetric 2-tensor S = S (g, h) is defined by
Sjk ~ 2gqp R;jk hrp - gqp Rjp hkq - gqp Rkp hjq.
Note that S involves no derivatives of h. The 1-form V may be rewritten as
Vi= ~gpq (\lphqk + 'Vqhpk - \lkhpq) = gPq9kr [D (I' 9 ) (h)J;q,
where
D (r 9 ) : C^00 (S2T* Mn)_, C^00 (S2T* Mn® T Mn)
denotes the linearization of the Levi-Civita connection and is given by[D (r g) (h)Jt = :s ls=o r~j
when- a I g =h.
OS s=O