their gradients and upper bounds of their Hessians on complete Riemannian
manifolds with bounded curvature. These distance-like functions are used
to construct the barrier functions referred to in the previous paragraph.
Moreover, this construction carries over to the case of the Ricci flow.
The aforementioned mollifiers are not only applicable to the proof of the
compactness theorem in regards to the center of mass (discussed in Chap-
ter 4 of Part I of this volume), but also to constructing a barrier function
used in the proof of Hamilton's matrix Harnack estimate in Chapter 15
on complete noncompact Riemannian manifolds with bounded nonnegative
curvature operator.
A fundamental property of the heat equation is that a solution which is
initially nonnegative immediately becomes either positive or identically zero.
This property is known as the strong maximum principle. For a solution
to the Ricci flow, the curvature tensor satisfies a heat-type equation. The
strong maximum principle for systems, due to Hamilton based on earlier
work of Weinberger and others, tells us that for solutions to the Ricci flow
with nonnegative curvature operator (such as singularity models in dimen-
sion 3) the curvature operator has a special form after the initial time. In
particular, the image of the curvature operator is independent of time and
invariant under parallel translation in space. Moreover, using the natural
Lie algebra structure on the fibers A^, x G M, of the bundle of 2-forms, the
image of the curvature operator is a Lie subalgebra. It is useful to observe
that the strong maximum principle is a local result and does not require the
solution to be complete or to have bounded curvature. We also formulate
the strong maximum principle in the more general setting of Chapter 10,
i.e., for sections of a vector bundle which solve a PDE.
In dimension 3, A^ is isomorphic to 50 (3) and its only proper nontrivial
Lie subalgebras are isomorphic to 50 (2), which is 1-dimensional. Hence,
after the initial time, a solution to the Ricci flow on a 3-manifold with non-
negative sectional curvature either is flat, has positive sectional curvature,
or admits a global parallel 2-form. In the last case, by taking the dual of this
2-form, we have a parallel 1-form and the solution splits locally as the prod-
uct of a surface solution and a line. This classification is useful in studying
the singularities which arise in dimension 3.
Chapter 13. In this chapter we discuss the following two topics in Ricci
(1) Curvature conditions that are not preserved. The weak positivity
(i.e., nonnegativity) of the following curvatures are preserved: scalar curva-
ture, Ricci curvature in dimension 3, Riemann curvature operator, isotropic
curvature, complex sectional curvature, as well as the 2-nonnegativity of
the curvature operator. On the other hand, in this chapter we discuss some
nonnegativity conditions which are not preserved under the Ricci flow. In
particular, we consider the conditions of nonnegative Ricci curvature and
nonnegative sectional curvature in dimensions at least 4 for solutions of the
Ricci flow on both closed and noncompact manifolds.
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