Heat kernel on smooth metric measure spaces with nonnegative curvature

Abstract

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry–Émery Ricci curvature. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-uniqueness property for nonnegative solutions of the $f$-heat equation, assuming $f$ is of at most quadratic growth. In particular, any $L_f^1$-integrable $f$-subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit $f$-heat kernel for Gaussian solitons.

Appendix

In the appendix we solve for the $f$-heat kernel of $1$-dimensional steady Gaussian soliton $(\mathbb {R},\ g_0, e^{-f}dx)$, where $g_0$ is the Euclidean metric, and $f=k x$ with $k=\pm 1$. The method is standard separation of variables. Suppose the $f$-heat kernel is of the form

$$\begin{aligned} H(x,y,t)= \varphi (y)\phi (x)\psi (t)\times \exp \left( -\frac{|x-y|^2}{4t}\right) . \end{aligned}$$

For a fixed $y$, we get

$$\begin{aligned} \begin{aligned} H_t&= \varphi \phi e^{-\frac{|x-y|^2}{4t}}\left( \psi _t+\psi \frac{|x-y|^2}{4t^2}\right) ,\\ H_x&= \varphi \psi e^{-\frac{|x-y|^2}{4t}}\left( \phi _x-\phi \frac{x-y}{2t}\right) ,\\ H_{xx}&= \varphi \psi e^{-\frac{|x-y|^2}{4t}} \left( \phi _{xx}+\phi \frac{|x-y|^2}{4t^2}-\phi _x\frac{x-y}{t}-\phi \frac{1}{2t}\right) . \end{aligned} \end{aligned}$$

So $H_t=H_{xx}-f_xH_x$ implies

$$\begin{aligned} \begin{aligned} \phi \left( \psi _t+\psi \frac{|x-y|^2}{4t^2}\right)&= \psi \left( \phi _{xx}+\phi \frac{|x-y|^2}{4t^2}-\phi _x\frac{x-y}{t}-\frac{\phi }{2t}\right) \nonumber \\&\quad -k\psi \left( \phi _x-\phi \frac{x-y}{2t}\right) . \end{aligned} \end{aligned}$$

That is,

$$\begin{aligned} \frac{\psi _t}{\psi }= \frac{\phi _{xx}-k\phi _x}{\phi }-\frac{x-y}{2t}\cdot \frac{2\phi _x-k\phi }{\phi }-\frac{1}{2t}. \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned} \frac{\phi _{xx}-k\phi _x}{\phi }&= C_1, \frac{(2\phi _x-k\phi )(x-y)}{\phi } = C_2, \frac{\psi _t}{\psi } = C_1-\frac{1+C_2}{2t}, \end{aligned} \end{aligned}$$

From above, their solutions are

$$\begin{aligned} \begin{aligned} \phi&= C_3 e^{\frac{1}{2}kx},\\ \psi&= C_4 \frac{1}{\sqrt{t}}e^{-4/t}, \end{aligned} \end{aligned}$$

where $C_1$, $C_2$, $C_3$, $C_4$ are constants.

By the initial condition $\lim _{t\rightarrow 0}u(x,t)=\delta _{f,y}(x)$ we get $\varphi (y)= e^{\frac{1}{2}ky}$, and $C_3C_4=\frac{1}{2\sqrt{\pi }}$. Therefore the $f$-heat kernel is

$$\begin{aligned} H(x,y,t)=\frac{e^{\pm \frac{x+y}{2}}\cdot e^{-t/4}}{(4\pi t)^{1/2}} \times \exp \left( -\frac{|x-y|^2}{4t}\right) . \end{aligned}$$

It is easy to check that $\int _{\mathbb {R}}H(x,y,t)e^{-f(x)}dx=1$, which confirms the stochastic completeness proved in Lemma 4.1.$\square$