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NOTES AND COMMENTARY 103

conclude that u is st rictly positive for all positive time. This is an example of
a strong maximum principle. The availability of strong maximum principles
for parabolic equations is one of the principal advantages of using evolution
equations in geometry. Heuristically, strong maximum principles show that
a pointwise bound which holds at some time t can propagate to a global
bound at any time t + c.
Strong maximum principles apply to both scalar and tensor quantities
evolving under the Ricci flow. We will study their general formulation in
the next volume. But we shall benefit from them in subsequent chapters
of this monograph. For example, if one has a solution of the Ricci flow
with initially nonnegative scalar curvature, the strong maximum principle

for scalar equations implies that R > 0 for all times t > 0 that the solution

exists - unless all g (t) are scalar fiat metrics. (See for instance Section 7 of
Chapter 5.) The strong maximum principle for tensors is also highly useful.
We shall sketch a key example here; rigorous arguments will appear in the
chapters which follow. If (M^3 ,g (t)) is a solution of the Ricci flow with

nonnegative sectional curvature at t = 0, then the eigenvalues v ::; μ :S A of

the curvature operator satisfy exactly one of the following patterns

O=v=μ=.A

O=v=μ<.A

0 < v ::; μ ::; .A

for all points and times t > 0 such that the solution exists. (The Lie algebra

structure of so (3) rules out the case 0 = v < μ :S .A.) In the first case,

the manifold is fiat; in the second case, it splits as the product of a one-

dimensional factor with a positively curved surface; and in the third case,

the manifold has strictly positive sectional curvature everywhere. Because

(as we shall see in Corollary 9.7) any solution of the Ricci flow constructed

as a limit of parabolic dilations about a finite-time singularity in dimension

n = 3 has nonnegative sectional curvature, this rigidity - a consequence

of the strong maximum principle for tensors - is very important for the

analysis of singularities.

Notes and commentary

Maximum principles (weak and strong) are among the most important

technical tools used to study the Ricci flow. They have numerous applica-

tions. For example, the scalar maximum principle implies that the bound

R 2:: Rmin (0) for the scalar curvature is preserved (Lemma 6.8). The ten-

sor maximum principle implies that nonnegative Ricci curvature and hence

nonnegative sectional curvature is preserved in dimension n = 3 (Corollary

6.11) and that nonnegativity of the curvature operator is preserved in all

dimensions (Corollary 6.27).