2.The maximum principle for heat-type equations

The maximum principle#

Proposition: (Maximum principle for supersolutions of the heat equation). Let $g(t)$ be a family of metrics on a closed manifold $M^{n}$ and let $u: M^{n} \times[0, T) \rightarrow \mathbb{R}$ satisfy

Then if $u \geq c$ at $t=0$ for some $c \in \mathbb{R}$, then $u \geq c$ for all $t \geq 0$.

Proof：

Corollary : (Lower bound of scalar curvature is preserved under RF). If $g(t), t \in[0, T)$, is a solution to the Ricci flow on a closed manifold with $R \geq c$ at $t=0$ for some $c \in \mathbb{R}$, then

for all $t \in[0, T)$. In particular, nonnegative (positive) scalar curvature is preserved under the Ricci flow.

Proof:

Lemma:(Maximum principle comparing with the ODE). Suppose $g(t)$ is a family of metrics on a closed manifold $M^{n}$ and $u: M^{n} \times[0, T) \rightarrow \mathbb{R}$ satisfies

where $X(t)$ is a time-dependent vector field and $F$ is a Lipschitz function. If $u \leq c$ at $t=0$ for some $c \in \mathbb{R}$, then $u(x, t) \leq U(t)$ for all $x \in M^{n}$ and $t \geq 0$, where $U(t)$ is the solution to the ODE

with $U(0)=c$.

Proof:

evolution equation for the scalar curvature#

Let $\left(M^{n}, g(t)\right), t \in[0, T)$, be a solution to the Ricci flow on a closed manifold Since $|\mathrm{Rc}|^{2} \geq \frac{1}{n} R^{2}$ , equation (2.2) implies

Since the solutions to the ODE $\frac{d \rho}{d t}=\frac{2}{n} \rho^{2}$ are $\rho(t)=\frac{n}{n \rho(0)^{-1}-2 t}$, by the maximum principle one has

for all $x \in M^{n}$ and $t \geq 0$. It $\rho(0)>0$, then $\rho(t)$ tends to infinity in finite time. When $M$ is closed, let $R_{\text {min }}(0) \doteqdot \inf _{t=0} R$.

Corollary:(Finite singularity time for positive scalar curvature). If $\left(M^{n}, g_{0}\right)$ is a closed Riemannian manifold with positive scalar curvature, then for any solution $g(t), t \in[0, T)$, to the Ricci flow with $g(0)=g_{0}$ we have

Definition: (Complete solution). A solution $g(t), t \in \mathcal{I}$, of the Ricci flow is said to be complete if for each $t \in \mathcal{I}$, the Riemannian metric $g(t)$ is complete.

Lemma (Ancient solutions have nonnegative scalar curvature). If $\left(M^{n}, g(t)\right), t \in(-\infty, 0]$, is a complete solution to the Ricci flow with bounded curvature on compact time intervals, then either $R(g(t))>0$ for all $t \in$ $(-\infty, 0]$ or $\operatorname{Rc}(g(t)) \equiv 0$ for all $t \in(-\infty, 0]$.

Proof:

1. EX1 Let $u$ be a solution to the heat equation with respect to a metric $g(t)$ evolving by the Ricci flow:

   $$\frac{\partial u}{\partial t}=\Delta_{g(t)} u$$

   Recall from Exercise 1.40 that

   $$\frac{\partial}{\partial t}|\nabla u|^{2}=\Delta|\nabla u|^{2}-2|\nabla \nabla u|^{2}$$
   • From this deduce
   $$\left(\frac{\partial}{\partial t}-\Delta\right)\left(t|\nabla u|^{2}+\frac{1}{2} u^{2}\right) \leq 0$$
   • Apply the maximum principle to conclude that if $M^{n}$ is closed, then
   $$\begin{equation*} |\nabla u| \leq \frac{U}{\sqrt{2} t^{1 / 2}} \tag{2.10} \end{equation*}$$

   where $U \doteqdot \max _{t=0}|u|$.

2. Ex2 Let $u$ be a solution to the heat equation on a Riemannian manifold $\left(M^{n}, g\right)$. Show that

   $$\frac{\partial}{\partial t}|\nabla u|^{2}=\Delta|\nabla u|^{2}-2|\nabla \nabla u|^{2}-2 R_{i j} \nabla_{i} u \nabla_{j} u$$

   What estimate, analogous to the previous exercise, for $|\nabla u|$ do you get assuming $\mathrm{Rc} \geq 0$ ? How about the case where $\mathrm{Rc} \geq-(n-1) K$ for some constant $K$ ? Which curvature condition do you need to get decay of $|\nabla u|$ as $t \rightarrow \infty$ ?