Properties of the solutions of the conjugate heat equation

2006/01/17 by Hamilton, Richard, Sesum, Natasa · 1 citation
#53C44 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0601415

Abstract

In this paper we consider the class A of those solutions u(x,t) to the conjugate heat equation (d)/(dt)u = -Δu + Ru on compact Kähler manifolds M with c₁ > 0 (where g(t) changes by the unnormalized Kähler Ricci flow, blowing up at T < ∞), which satisfy Perelman's differential Harnack inequality on [0,T). We show A is nonempty. If |\ric(g(t))| ≤ (C)/(T-t), which is alaways true if we have type I singularity, we prove the solution u(x,t) satisfies the elliptic type Harnack inequlity, with the constants that are uniform in time. If the flow g(t) has a type I singularity at T, then A has excatly one element.

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• Backward Harnack inequality and Hamilton estimates for heat type equations

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