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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 ;-))

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  • $\begingroup$ No one here, who can help me? $\endgroup$
    – d.s.
    May 7, 2016 at 7:36

1 Answer 1

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I hope somebody has already answered you and that you didn't get stuck too long in these questions. Anyway, for your first question, take any geodesic $\gamma$ parametrized by arc-length between $x_{1}$ and $x_{2}$. Then

\begin{eqnarray*} |f(x_{1})-f(x_{2})| & = & |f\circ \gamma (0) - f \circ \gamma(l) | \\ & = & |\int_{0}^l (f \circ \gamma)'(t) dt | \\ & \le & \int_{0}^l |(f \circ \gamma)'(t)| dt \\ & \le & \int_{0}^l |<\nabla f (\gamma (t)), \gamma'(t)>| dt \end{eqnarray*} whence the result by successively using Cauchy-Schwarz, the fact that $|\gamma'(t)|=1$ a.e. as $\gamma$ is parametrized by arc-length, and taking the infimum.

For the second question right now I don' know, but I think about it and I'll try to bring you an answer later.

For your third question, you could use the Rayleigh quotient characterisation of $\lambda_{1}$: $$ \lambda_{1} = \inf_{f \in H^1_{0} (B)} \frac{\int_{B} |\nabla f|^2 dVol }{\int_{B} |f|^2 dVol}. $$ The Poincaré inequality gives you that for any $f \in H^1_{0}(B)$, $$ \frac{\int_{B} |\nabla f|^2 dVol }{\int_{B} |f|^2 dVol} \ge \frac{1}{Cr^2}.$$ So if I'm not mistaking you should obtain $\lambda_{1} \ge \frac{1}{Cr^2}$.

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