I'm trying to work through the 2nd chapter "Volume comparison" of Jeff Cheeger's "Degeneration of Riemannian Metrics under Ricci Curvature Bounds". But with the last section I have some problems.
First some notation: Let $ M^n $ be a Riemannian Manifold, $ p\in M^n $. $ \mathcal A(r) $ denotes the area element of $ \partial B_r(p) $ in geodesic polar coordinates. Symbols, that are underlined, are in the complete simply connected space with constant curvature $ H $ ($H$ is a real constant).
For $h\geq 0$ a nonnegative function, we define $$ \mathcal F_h(x_1,x_2)=\inf_{\gamma} \int_0^l \! h(\gamma(s)) \, ds $$ (here the infimum is taken over all minimal geodesics from $x_1$ to $x_2$, $s$ denotes arclength.
Cheeger is proving the segment inequality: Let $ \operatorname{Ric}_{M^n}\geq -(n-1) $ and $ A_1, A_2\subseteq B_r(p) $ with $ r\leq R $. Then $$ \int_{A_1\times A_2}\! \mathcal F_h(x_1,x_2) \, d(x_1,x_2)\leq c(n,R) r (\operatorname{vol}(A_1)+\operatorname{vol}(A_2))\cdot \int_{B_{2R}(p)}\! h(x) \, dx, $$ where $$ c(n,R)=\sup_{0<\frac s2\leq u\leq s} \frac{\underline{\mathcal A}(s)}{\underline{\mathcal A}(u)}. $$
Now to my problems: In the next section about the Poincaré inequality, Cheeger writes: If $h=|\nabla f|$, then $|f(x_1)-f(x_2)|\leq \mathcal F_h(x_1, x_2)\quad (*)$. Has anyone some hints, how to show this?
In the next paragraph, he shows a lower bound for the Dirichlet problem for $B_r(p)$: Let $\operatorname{Ric}_{M^n}\geq (n-1)H$ and $\partial B_{3r}(p)\neq \emptyset$. If $f: B_r(p)\to\mathbb R$ with $f|_{\partial B_r(p)}=0$, we can extend $f$ to $B_{3r}(p)$ by setting $f=0$ in $B_{3r}(p)\setminus B_r(p)$. If we choose $h=|\nabla f^2|$, a straightforward application of the segment inequality and relative volume comparison shows, that there exists $x_1\in B_{\frac 32 r}(p)\setminus B_r(p)$, such that $$ \int_{B_{\frac 32 r(p)}}\! |f^2(x_2)|\, dx_2\leq \int_{B_{\frac 32 r(p)}}\! |\mathcal F_{|\nabla f^2|}(x_1, x_2)|\, dx_2\leq c(n,R)r \int_{B_{\frac 32 r(p)}}\! |\nabla f^2(x_2)|\, dx_2. $$
Here is my next problem: The first inequality follows from $(*)$. But I have no idea, how to show the second inequality.
And this should "easily imply", that $ \lambda_1\geq c(n,R)r^2>0 $ (where $\lambda_1$ is the smallest eigenvalue for the Dirichlet problem for $B_r(p)$). Same here: Some hints to show this? (I think, "easily imply" is too difficult for me ;-))
Thank you very much for some help. (And I hope my English is not too bad ;-))