CHAPTER 10 Weak Maximum Principles for Scalars, Tensors, and Systems And if you go chasing rabbits ... — From "White Rabbit" by Grace Slick of Jefferson Airplane In this chapter we present (starting in Section 2) the statements and proofs of various versions of the weak maximum principle for systems. We restrict ourselves to closed manifolds and in Chapter 12 we discuss the com- plete noncompact case. The preliminary formulation we give is for sections of vector bundles solving heat-type equations where the Riemannian metrics on the base manifold and the connections on the vector bundle are time- dependent (for example, the metrics solve the Ricci flow equation) whereas the metric on the fibers is fixed. In the study of the Ricci flow we are able to reduce to the fixed fiber-metric case by Uhlenbeck's trick (see Section 2 of Chapter 6 in Volume One). In a sense, all proofs of maximum principles for classical solutions boil down to the second derivative test in calculus. The main application in the Ricci flow has been to the study of the evolu- tion of the Riemann curvature tensor.1 Another application is to the proof of Hamilton's matrix Harnack estimate for the Ricci flow (see Section 4 of Chapter 15). We first review the statements and some applications of the weak max- imum principles for scalars and tensors which were discussed in Volume One. Then we give proofs of the Weinberger-Hamilton maximum principle for systems including some generalized versions, e.g., time-dependent fiber- wise convex sets and the avoidance maximum principle. Regarding the case where the fiberwise convex sets depend continuously on time, in subsection 3.4 we give Bohm and Wilking's simplified proof. As applications of the aforementioned maximum principles, in this chapter we briefly survey the following results, most of which were proved in Volume One and are due to Hamilton.2 Applications of the scalar maximum principles are (1) lower bound for the scalar curvature, Many of the maximum principles in this chapter also apply to other geometric flows such as the mean curvature flow. Independently by Ivey in the case of the Hamilton-Ivey estimate. For the classific- cation of 4-manifolds with positive isotropic curvature there is work of Chen-Zhu [110] on fixing a gap in Hamilton's proof partially using Perelman's work. 1 http://dx.doi.org/10.1090/surv/144/01

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