What are the known relations between isoperimetric and Poincaré inequalities on manifolds?


For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the isoperimetric constant to the first eigenvalue of the Laplacian. Does this imply a Poincaré inequality for such manifolds?



Isoperimetry and Poincaré Inequality
Everything here is for closed Riemannian manifolds.
If you have a lower bound on Cheeger's isoperimetric constant $h(M)$, then Cheeger's inequality
$\lambda\_1(M)\geq \frac{h(M)^2}{4}$ gives you a lower bound on the first eigenvalue of the Laplacian.
Thanks to the variational characterization of $\lambda\_1(M)$, this is exactly equivalent to an upper bound on the constant $C\_P$ in the $L^2$ Poincaré inequality
$$\int\_M f^2 dV \leq C\_P \int\_M |\nabla f|^2 dV,$$
for all smooth functions $f$ with $\int\_M f dV=0$.


If you have a lower bound for the Ricci curvature, and also an upper bound for $C\_P$ (equivalently, a lower bound for $\lambda\_1(M)$), then Buser's inequality gives you a lower bound for $h(M)$.


See for example [Wikipedia's page.](http://en.wikipedia.org/wiki/Cheeger_constant)


On the other hand [a result of Yau](http://www.numdam.org/numdam-bin/item?id=ASENS_1975_4_8_4_487_0) shows that $h(M)$ is equal to the reciprocal of the constant $C\_P'$ in the $L^1$ Poincaré inequality
$$\int\_M |f| dV \leq C\_P' \int\_M |\nabla f| dV,$$
for all smooth functions $f$ with $\int\_M f dV=0$.


If you are interested in obtaining bounds on $h(M)$ or $\lambda\_1(M)$ in terms of geometric quantities, there are the following basic results: $h(M)$ can be bounded below in terms of a lower bound for the volume of $M$, an upper bound for its diameter, and a lower bound for its Ricci curvature ([Croke](http://www.numdam.org/numdam-bin/item?id=ASENS_1980_4_13_4_419_0)). On the other hand $\lambda\_1(M)$ can be bounded below in terms of an upper bound for its diameter and a lower bound for its Ricci curvature ([Li-Yau](http://www.ams.org/mathscinet-getitem?mr=0573435)). Of course these bounds also depend on the dimension of the manifold.


Finally, if you are intersted in the relation with Sobolev inequalities, you should look at [this paper of Peter Li.](http://www.numdam.org/numdam-bin/item?id=ASENS_1980_4_13_4_451_0)


