112 5. THE RICCI FLOW ON SURFACES
solitons. In particular, we shall obtain upper bounds for the scalar curvature
by estimating such a quantity. These upper bounds are uniform except when
r > 0, in which case they are exponential. (Ricci solitons and self-similar
solutions were introduced briefly in Section 1 of Chapter 2. In a chapter of
the planned successor to this volume, we will make a more systematic study
of Ricci solitons in all dimensions.)
DEFINITION 5.10. A solution g(t) of the normalized Ricci fl.ow on a
surface M^2 is called a self-similar solution of the normalized Ricci fl.ow if
there exists a one-parameter family of conformal diffeomorphisms rp(t) such
that
g(t) = rp(t)*g(O).Differentiating this equation with respect to time implies that(5.5)[)at 9 = £xg,
where X(t) is theone-parameter family of vector fields generated by rp(t).
By definition of the normalized Ricci flow, equation (5.5) is equivalent to
(5.6)If X = -V' f for some function f ( x, t), we obtain
(5.7)DEFINITION 5.11. We say a metric g (t) on a surface M^2 is a Ricci
soliton if it satisfies equation (5.6). We say g (t) is a gradient Ricci
soliton if it satisfies equation ( 5. 7).
Tracing (5.7) yields the equation(5.8) !:lf = R - r.
This equation is solvable even on non-soliton solutions, because
{ (R - r) dA = 0.
}M2Thus for any solution g (t) of the normalized Ricci flow, the solution f of
(5.8) is defined to be the potential of the curvature. On a compact
(closed) surface M^2 , the potential is unique up to addition of function c (t)
of time alone, because the only harmonic functions there are constants.
Denoting the trace-free part of the Hessian of the potential f of the
curvature by
(5.9)we observe that the gradient Ricci soliton equation (5. 7) is equivalent to
M = 0. Recalling that Rik = ~9ik and taking the divergence of M,· we