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. xvii

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