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 ;-))

  • $\begingroup$ No one here, who can help me? $\endgroup$ – d.s. May 7 '16 at 7:36

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 $\lambda_{1} \ge \frac{1}{Cr^2}$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.