Contents of Part II of Volume Two
Councillor Hamann: Almost no one comes down here, unless, of course, there's
- From the movie "The Matrix Reloaded" by the Wachowski brothers.
Chapter 10. In this chapter we discuss the general formulation of the
weak maximum principle for systems on closed manifolds, which applies to
bilinear forms such as curvature tensors. The maximum principle for scalars
may be considered as stating that solutions to a semilinear PDE are bounded
by the solutions to the associated ODE obtained by dropping the Laplacian
and any gradient terms. In particular for subsolutions/supersolutions to
the heat equation, the maximum/minimum is nonincreasing/nondecreasing.
This last statement has a generalization to symmetric 2-tensors, considered
in Chapter 4 of Volume One, which gives general sufficient conditions to
prove that the nonnegativity of tensor supersolutions to heat-type equations
is preserved. We had previously applied this to the Ricci tensor and also
obtained pinching estimates for the curvatures this way.
To obtain various estimates for the curvatures in Volume One, we found
it convenient to employ a more general formulation of the weak maximum
principle. We prove this version in this chapter. More precisely we consider
sections of vector bundles which satisfy a semilinear heat-type equation.
The maximum principle for systems states that if the initial section lies in
a subset of the vector bundle which is convex in the fibers and invariant
under parallel translation and if the associated ODE obtained by dropping
the Laplacian preserves this subset, then the solution to the PDE stays inside
this convex set.
The idea of the proof of this maximum principle is as follows. One
can prove the maximum principle for functions by considering the spatial
maximum function, which is Lipschitz in time, and showing that it is nonin-
creasing for subsolutions to the heat equation. In the case of the maximum
principle for systems, one can look at the function of time which is the maxi-
mum distance of the solution to the PDE from the subset. Using the support
functions to the convex fibers, one can show that this maximum distance
function s (£), which is again Lipschitz in time, satisfies an ODE of the form
ds/dt Cs. Since s (0) = 0, we conclude that s (t) = 0 and the maximum
principle for systems follows.