Refinements of the maximum principle include the case when the sub-
sets with convex fibers are time-dependent and also when there is a so-called
avoidance set for the solutions of the PDE. In terms of applications, one of
the most important special cases is when the sections of the bundle are bi-
linear forms. We discuss this case and applications to the curvature tensors.
We also discuss the Aleksandrov-Bakelman-Pucci maximum principle for
elliptic equations.
Chapter 11. In this chapter we present Bohm and Wilking's solu-
tion to the conjecture of Rauch and Hamilton on the classification of closed
Riemannian manifolds with positive curvature
The flavor of this
chapter is more algebraic with an essential component of the proof being the
irreducible decomposition of (algebraic) curvature tensors. Generally, one
of the ideas is to study when linear transformations of convex preserved sets
(with respect to the ODE corresponding to the PDE satisfied by Rm) remain
preserved. More specifically, via a 1-parameter family of linear transforma-
tions the cone of 2-nonnegative curvature operators is mapped into the cone
of nonnegative curvature operators. This reduces the classification prob-
lem for Riemannian manifolds with 2-positive curvature operator to that for
manifolds with positive curvature operator. To study the latter problem,
one would hope that the cones of 2-positive curvature operators with arbi-
trary Ricci pinching are preserved. Unfortunately there does not seem to
be any known way to prove this. Instead, one can prove that suitable linear
transformations of the cones of 2-positive curvature operators with arbitrary
Ricci pinching are preserved. This is sufficient to prove the Rauch-Hamilton
Chapter 12. This chapter comprises two main topics: (1) weak max-
imum principles on noncompact manifolds and (2) the strong maximum
principle, which is a local result. In both cases we consider scalar parabolic
equations and systems of parabolic equations.
Since singularity models are often noncompact, it is important to be
able to apply the weak maximum principle on complete, noncompact man-
ifolds. We begin with the heat equation and present the weak maximum
principle of Karp and Li which applies to solutions with growth slower than
exponential quadratic in distance. Using barrier functions, we then give a
weak maximum principle for bounded solutions of heat-type systems. As a
special case, we show that complete solutions to the Ricci flow on noncom-
pact manifolds, with nonnegative curvature operator initially and bounded
curvature on space and time, have nonnegative curvature operator for all
We also discuss mollifiers on Riemannian manifolds with a lower bound
on the injectivity radius. One technical issue we discuss, using the mollifiers
obtained above, is the construction of distance-like functions with bounds on
They actually obtain the stronger result of classifying closed manifolds with 2-
positive curvature operator.
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