Contents of Part II of Volume Two

Councillor Hamann: Almost no one comes down here, unless, of course, there's

a problem.

- 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.

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