Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Question

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ $$\left(\int_M |u-\overline{u}|^p\right)^{\frac 1p} \leq C\left(\int_M |\nabla u|^q\right)^{\frac 1q}.$$ I'm not interested in the best constant $C$ but want to know how does $C$ depend on $M$? I am hoping for something like $C=C(|M|,n)$ where $|M|$ is the volume of $M$. Can someone refer me to this result? Thanks.

Answer 1

No, it's not nearly that simple. The constant $C$ quantifies the connectivity of the manifold. It can be imagined as the severity of traffic jams that occur when all inhabitants of the manifold decide to drive to a random place at the same time. For example, let $M$ be two unit spheres $S^2$ joined by a thin cylinder of radius $r\ll 1$ and length $1$. There is a smooth function $u$ that is equal to $1$ on one sphere, $-1$ on the other, and has gradient $\approx 2$ on the cylinder. For this function $\bar u=0$, so $$\left(\int_M |u-\overline{u}|^p\right)^{\frac 1p} \approx (8\pi)^{1/p} $$ On the other hand, the gradient is suppored on the cylinder, so $$\left(\int_M |\nabla u|^q\right)^{\frac 1q} \approx (4\pi r)^{1/q} $$ The constant $C$ depends on $r$ and blows up as $r\to 0$, though the diameter and volume of the manifold hardly change.

See also: Poincaré Inequality.