1547671870-The_Ricci_Flow__Chow

1. THE EVOLUTION OF CURVATURE

(5) The volume form dμ of g evolves by

a 1

~dμ = - (tr 9 h) dμ.

ut 2

Substituting h = -2 Re into the lemma yields the following result.

COROLLARY 6.6. Suppose g (t) is a solution of the Ricci flow.

(6.1)

(6.2)

(6.3)

(6.4)

(6.5)

(1) The Levi-Civita connection r of g evolves by

atrij a k = -g k£ C'viRje + \ljRie - \leRij).

(2) The (3, 1 )-Riemann curvature tensor Rm of g evolves by

a { -\liY'jRkp - \li\lkRjp + \li\lpRjk }

- at Re.k i] = g£P

+\lj\liRkp + \lj\lk~p - \lj\lpRik

(3) The Ricci tensor Re of g evolves by

a

atRjk = b..Rjk + \lj\lkR-gPq (\lq\ljRkp + \lq\lkRjp)

(4) The scalar curvature R of g evolves by

at a R = 2b..R - 2g^1 "k gPq\J q \ljRkp + 2 IRcl^2

(5) The volume form dμ of g evolves by

a

atdμ =-Rdμ.

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We shall soon obtain more useful forms of many of the equations above
by applying various curvature identities. To motivate the type of equation we
are looking for, recall formula (5.3) for the evolution of the scalar curvature
of a solution of the normalized Ricci fl.ow on a surface:

a

atR=b..R+R(R- r).

This is an example of a reaction-diffusion equation with a quadratic

nonlinearity involving the scalar curvature. Equations of this form are also

called heat-type equations. We shall see below that in any dimension

n ~ 2, the Riemann curvature Rm, the Ricci curvature Re, and the scalar

curvature Rall satisfy heat-type equations whose nonlinear terms are various

quadratic contractions of the Riemann curvature.

1.1. The scalar curvature function. We first consider the scalar

curvature function, since its evolution equation is the simplest. After two

contractions of the second Bianchi identity

\7 mRijk£ + \7 kRij£m + \7 eRijmk = 0,

one gets gJk\ljRke = ~\leR. Applying this to (6.4) gives a nicer equation.