Monotonicity formulas via the Bakry-Émery curvature

We prove several monotone formulas for smooth metric measure spaces with nonnegative m-Bakry-Émery curvature. First, we define Af(r) and Vf(r) for the weighted Green function, and establish the monotone formulas. These formulas are parallel to the weighted volume comparison theorem on smooth metric measure spaces with nonnegativem-Bakry-Émery curvature. Second, we define the weighted Nash entropy, Fisher information and Perelman's entropy. Then we get the monotone formulas along the weighted heat equation via the m-Bakry-Émery curvature.