Differential Harnack inequality for solutions of the conjugate heat equation

Proposition 9.1   Let $g_{ij}(t)$ be a solution to the Ricci flow $(g_{ij})_t=-2R_{ij}, 0\le t\le T,$ and let $u=(4\pi(T-t))^{-\frac{n}{2}}e^{-f}$ satisfy the conjugate heat equation $\Box^*u=-u_t-\triangle u+Ru=0.$ Then $v=[(T-t)(2\triangle f-\vert\nabla f\vert^2+R)+f-n]u$ satisfies

$$\Box^*v=-2(T-t)\vert R_{ij}+\nabla_i\nabla_j f-\frac{1}{2(T-t)}g_{ij}\vert^2$$ (9.1)

Proof. Routine computation.

Clearly, this proposition immediately implies the monotonicity formula (3.4); its advantage over (3.4) shows up when one has to work locally.

Corollary 9.2   Under the same assumptions, on a closed manifold $M$,or whenever the application of the maximum principle can be justified, min$\ v/u$ is nondecreasing in $t.$

Corollary 9.3   Under the same assumptions, if $u$ tends to a $\delta$-function as $t\to T,$ then $v\le 0$ for all $t<T.$

Proof. If $h$ satisfies the ordinary heat equation $h_t=\triangle h$ with respect to the evolving metric $g_{ij}(t),$ then we have $\frac{d}{dt}\int{hu}=0$ and $\frac{d}{dt}\int{hv}\ge 0.$ Thus we only need to check that for everywhere positive $h$ the limit of $\int{hv}$ as $t\to T$ is nonpositive. But it is easy to see, that this limit is in fact zero.

Corollary 9.4   Under assumptions of the previous corollary, for any smooth curve $\gamma(t)$ in $M$ holds

$$-\frac{d}{dt}f(\gamma(t),t)\le\frac{1}{2}(R(\gamma(t),t)+\vert\dot{\gamma}(t)\vert^2) -\frac{1}{2(T-t)}f(\gamma(t),t)$$ (9.2)

Proof. From the evolution equation $f_t=-\triangle f+\vert\nabla f\vert^2-R+\frac{n}{2(T-t)}$ and $v\le 0$ we get $f_t+\frac{1}{2}R-\frac{1}{2}\vert\nabla f\vert^2-\frac{f}{2(T-t)}\ge 0.$ On the other hand, $-\frac{d}{dt}f(\gamma(t),t)=-f_t-<\nabla f,\dot{\gamma}(t)>\le -f_t+\frac{1}{2}\vert\nabla f\vert^2+\frac{1}{2}\vert\dot{\gamma}\vert^2.$ Summing these two inequalities, we get (9.2).

Corollary 9.5   If under assumptions of the previous corollary, $p$ is the point where the limit $\delta$-function is concentrated, then $f(q,t)\le l(q,T-t),$ where $l$ is the reduced distance, defined in 7.1, using $p$ and $\tau(t)=T-t.$

Proof. Use (7.13) in the form $\Box^*$exp$(-l)\le 0.$

9.6 Remark. Ricci flow can be characterized among all other evolution equations by the infinitesimal behavior of the fundamental solutions of the conjugate heat equation. Namely, suppose we have a riemannian metric $g_{ij}(t)$ evolving with time according to an equation $(g_{ij})_t=A_{ij}(t).$ Then we have the heat operator $\Box=\frac{\partial}{\partial t}-\triangle$ and its conjugate $\Box^*=-\frac{\partial}{\partial t}-\triangle-\frac{1}{2}A,$ so that $\frac{d}{dt}\int{uv}=\int{((\Box u)v-u(\Box^* v))}.$ (Here $A=g^{ij}A_{ij}$) Consider the fundamental solution $u=(-4\pi t)^{-\frac{n}{2}}e^{-f}$ for $\Box^*,$ starting as $\delta$-function at some point $(p,0).$ Then for general $A_{ij}$ the function $(\Box\bar{ f}+\frac{\bar{f}}{t})(q,t),$ where $\bar{f}=f-\int{fu},$ is of the order $O(1)$ for $(q,t)$ near $(p,0).$ The Ricci flow $A_{ij}=-2R_{ij}$ is characterized by the condition $(\Box\bar{ f}+\frac{\bar{f}}{t})(q,t)=o(1);$ in fact, it is $O(\vert pq\vert^2+\vert t\vert)$ in this case.

9.7* Inequalities of the type of (9.2) are known as differential Harnack inequalities; such inequality was proved by Li and Yau [L-Y] for the solutions of linear parabolic equations on riemannian manifolds. Hamilton [H 7,8] used differential Harnack inequalities for the solutions of backward heat equation on a manifold to prove monotonicity formulas for certain parabolic flows. A local monotonicity formula for mean curvature flow making use of solutions of backward heat equation was obtained by Ecker [E 2].