1547671870-The_Ricci_Flow__Chow

CHAPTER 3

Short time existence

A foundational step in the study of any system of evolutionary par-
tial differential equations is to show that it enjoys short-time existence and
uniqueness. In this chapter, we prove short-time existence of the Ricci fl.ow.
Because the fl.ow is quasilinear and only weakly parabolic, short-time exis-
tence does not follow from standard parabolic theory. Hamilton originally
used the sophisticated machinery of the Nash- Moser implicit function theo-
rem to establish short-time existence in [58]. Our proof follows the elegant
method suggested by Dennis DeTurck in [36].

1. Variation formulas
   Given any smooth family of metrics g (t) on a smooth manifold Mn, one
   may compute the variations of the Levi-Civita connection and its associated
   curvature tensors. These calculations will be applied later to derive the
   evolution equations for these quantities under the Ricci flow. (See Chapter
   6.) We compute the variation formulas now because the variation of the Ricci
   tensor determines the symbol of the Ricci flow equation when we regard the
   Ricci tensor Re (g) as a second order partial differential operator on the
   metric g. (These variation formulas may also be found in [20].)
   We first derive the variation formula for the Levi-Civita connection.
   Although a connection is not a tensor, the difference of two connections is
   a (2, 1)-tensor. In particular, the time-derivative of a connection is a (2, 1)-
   tensor. In this section, we shall assume that
   a

(3.1) 8t9ij = hij'

where h is some symmetric 2-tensor.

LEMMA 3.1. The metric inverse g -^1 evolves by

(3.2) :tgij = - gikgjehke.

PROOF. Since the Kronecker delta satisfies

equation (3.1) implies that

Sei = g ik 9ke,

(

0 = atl a k) 9ke + l ·k hke·

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