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Cheeger-Colding-Minicozzi: 1995 Linear growth harmonic functions on complete manifolds with nonnegative ricci curvature in GAFA Page 952: From Laplacian comparison, we have for $r<R$, \begin{equation*} \begin{aligned} 0 &=\int_{B_r(p)}\Delta h_R=\int_{\partial B_r(p)}\frac{\partial h_R}{\partial r}\\ &\ge \frac{\partial}{\partial r}\int_{\partial B_r(p)} h_R -\frac{n-1}{r}\int_{\partial B_r(p)} h_R \end{aligned} \end{equation*} where $h_R$ is a harmonic function on $B_R(p)\subset M^n$, $M^n$ is open Riemannian manifold with nonnegative Ricci curvature, and $0<r<R$. I don't know how to derive the inequality.

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If $J(r,\theta) dr \wedge d\theta$ is volume form then $\frac{\partial}{\partial r}J(r,\theta) = H J$ where $H$ is a mean curvature. If $M$ has a nonnegative sectional curvature, then $H \leq \frac{n-1}{r}$.

Whence $\frac{\partial}{\partial r}\int_{\partial B_r(P)} h_R \leq \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R + \int_{\partial B_r(p)} h_R \frac{\partial}{\partial r} J dr\wedge d\theta$ $ \leq \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R + \int_{\partial B_r(p)} h_R H J dr\wedge d\theta $

$\leq \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R + \frac{n-1}{r} \int_{\partial B_r(p)} h_R $

And $ \int_{ B_p(r)} \Delta h_R = \int_{\partial B_r(p)} \frac{\partial}{\partial r} h_R $ is a divergence theorem.

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