1547671870-The_Ricci_Flow__Chow

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by

9. TYPE I SINGULARITIES

p : JP r--+ min JP (V, V) ,

IVl=l

then v (JP) = p (JP) and

p(sJP+ (1-s)Q) 2 sp(JP) + (1-s)p(Q).

The fact that M remains in JC under the ODE (6.31) follows from (6.32),
because
d
-v = v^2 + >..μ > 0
dt
whenever v > 0. The lemma now follows from the maximum principle for
systems, which says that if Rm (x, 0) 2 0, then Rm (x, t) 2 0 for all t 2 0
such that the solution exists. D

The fact that positive sectional curvature is preserved in dimension 3 is a
special case of the more general result that positivity of the curvature oper-
ator is preserved in all dimensions. A result which is special to 3 dimensions
is the following:

LEMMA 9.3 (Positive Ricci curvature is preserved.). If (M^3 ,g(t)) is a

solution of the Ricci flow such that the initial metric g(O) has positive {non-

negative) Ricci curvature, then the metrics g(t) have positive {nonnegative)

Ricci curvature for all t > 0 that the solution exists.

PROOF. The proof is analogous to the previous lemma. Here we define

JC by

JC = {JP : v (JP) + μ (JP) > 0} '

so that the quadratic form M corresponding to Rm [g] lies in JC if and only
if Re [g] > 0. JC is fiberwise convex, because the function
v (JP)+μ (JP) = min [JP (V, V) +JP (W, W)]
IVl=IWl=l
(V,W)=O
is concave in each fiber. If v + μ > 0, we have >.. 2 μ 2 (v + μ) /2 > 0 and
hence
d

dt (v + μ) = v2 + >..μ + μ2 + >..v > >.. (v + μ) > 0.

The lemma follows again from the maximum principle for systems. D

3. Positive sectional curvature dominates
   The Ricci flow prefers positive curvature, in the sense that it preserves
   positive curvature operator in any dimension. Moreover, we saw in Section
   2 that both positive sectional curvature and positive Ricci curvature are
   preserved in dimension 3. Thus it is natural to investigate the extent to
   which the sectional curvatures of an arbitrary initial metric tend to become
   positive under the flow. By the maximum principle, it suffices to study the
   corresponding system (6.32) of ODE. We are interested in the case that the
   smallest eigenvalue v is negative. To simplify and motivate the qualitative