Question

What are the known relations between isoperimetric and Poincare 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 Poincare inequality for such manifolds?

Answer 1

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.

On the other hand a result of Yau 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). 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). 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.