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
Is this Laplacian comparison?
1 Answers
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.