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Let $\mathbf{M}$ be a submanifold of $\mathbb{R}^n$ with the induced Euclidean metric, and $\mbox{Ricc} \geq - \kappa , \kappa \geq 0$, as well as diameter bounded by $D$.

What is the best known bound on the Poincare constant of $\mathbf{M}$? To be completely precise, what is the smallest $C$, s.t. for all smooth $f: \mathbf{M} \to \mathbb{R}$, $$\mbox{var}_{\mu}(f) \leq C \mathbb{E}_{\mu} \|\nabla f\|^2$$ where $\mu$ is the uniform distribution over $\mathbf{M}$.

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The optimal (in terms of diameter $D$) Sobolev-Poincare inequality for manifolds with Ricci bounded below was proved by P. Maheux, L. Saloff-Coste. From the Sobolev-Poincare inequality and Holder's inequality you can deduce Poincare inequality. For details see: https://mathoverflow.net/a/321012/121665. I hope it helps.

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  • $\begingroup$ Is this known to be tight? In particular, is it know the dependence on $\kappa$ and $D$ needs to be exponential? $\endgroup$
    – Andy Mack
    Commented Feb 1, 2019 at 20:34
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    $\begingroup$ @AndyMack I do not know examples showing sharpness of the inequality, but as far as I know this is the best inequality known. I would suggest to search who quoted the paper of Maheux and Saloff-Coste to see if there are better results. $\endgroup$
    – Piotr Hajlasz
    Commented Feb 1, 2019 at 20:38
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    $\begingroup$ I think the exponential dependence might be realized for hyperbolic manifolds. $\endgroup$
    – Deane Yang
    Commented Feb 1, 2019 at 20:42
  • $\begingroup$ @DeaneYang: I don't quite see this (though I'm a newbie). Is it possible to do the calculation explicitly or something in this case? $\endgroup$
    – Andy Mack
    Commented Feb 1, 2019 at 22:50

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