Chapter 26. Bounds for the Heat Kernel for Evolving Metrics
Only love can bring the rain.- From "Love, Reign o'er Me" by The Who
In this chapter we discuss applications of the a priori estimates of the
previous chapter to obtain bounds for the heat kernel.
In §1 we discuss some basic general properties of solutions of the heat
equation.
In §2 we discuss the application of the mean value inequality and the
Li-Yau differential Harnack estimate, discussed in the previous chapter, to
obtain upper and lower bounds for heat kernels on evolving complete Rie-
mannian manifolds.
In §3 we present the space-time mean value property (MVP).
In §4 we discuss Tam's work, which is related to the. earlier works of
others, on the existence of distance-like functions with uniformly bounded
gradients and Hessians on complete noncompact manifolds with bounded
curvature.
1. Heat kernel for an evolving metric
In this section we discuss general properties of the heat equation with
respect to an evolving metric and consider the adjoint heat equation and its
associated heat kernel. In particular we consider Duhamel's principle, the
boundedness of the L^1 -norm, and the semigroup property.
1.1. The heat equation and its adjoint.
Let g ( r), r E [O, T], be a smooth family of Riemannian metrics on amanifold Mn and define
(26.1)
where Rij is a time-dependent symmetric 2-tensor. The application we shall
later make is to the case where Rij = Rij is the Ricci tensor. Let
(26.2)so that
(26.3)R _, -;-g ijR ij'
where dμg(r) denotes the volume form of g (r).
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