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By a theorem of Lichnerowicz, on a riemannian manifold $M^{(m)}$ with positive Ricci curvature, the reciprocal of Sobolev constant(ie. the first eigenvalue of laplacian) can be bounded from below by the quantity $inf |Ric|$. It has been generalised to bounding the Sobolev constant via $K=||Ric||_{L^p}$ by S.Gallot, wherein $p > \frac{m}{2}$. I assume that this holds even if the metric degenerates to a semi-positive metric on a subset of $M$ while keeping $K$ bounded. Theorem of Lichnerowicz does not, obviously, hold for degenerate metrics. My question is whether other geometric -sufficient- conditions are known that can be used to bound the Sobolev constant on manifolds with -possibly- degenerate metrics.

Edit: The Poincar\'e constant I consider is $C$ in the following: $||u - \bar{u}||_{2} $ $ \leq C ||\nabla u||_{2}$.

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Your first statement is not correct! Think about a family of flat tori $S^1\times S^1$ with product metrics where you make the length of one of the two $S^1$ factors (or both!) go to zero. The Sobolev constant in this case blows up. By results of Croke, Li-Yau and others, you can bound the Sobolev and Poincare' constants uniformly if you have a uniform lower bound for the Ricci curvature, a lower bound for the total volume, and a diameter upper bound. –  YangMills Jun 24 '12 at 0:33
    
Do you have a precise definition of "degenerate metric"? –  YangMills Jun 27 '12 at 19:36
    
By a degenerate metric, I mean a metric that might be semi-definite at some points. (Let's assume that this set is a no wehere dense subset). What probably is important is the `rate' of degeneration. One concrete example might be the following: let $g$ be a positive definite metric and consider $u g$, obtained by a conformal change, wherein $u$ is merely non-negative. Under what conditions on $u$, or geometric properties of the metric $u g$ can we guarantee a bound on Poincar\'e constant? By the way, I think the definition of Sobolev constant in your example is the reciprocal of mine. –  S.A.A Jun 27 '12 at 20:44
    
You haven't stated Gallot's theorem properly. –  Deane Yang Sep 26 '12 at 0:42
    
@Deane Yang: Do you mean the fact that it requires the $L^p$ bound of the negative part of the Ricci curvature? –  S.A.A Sep 26 '12 at 0:47
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I will comment only on the direction which I know of. That is, if you take the optimal mass transportation definition (by Lott-Villani and Sturm) of Ricci curvature lower bounds in metric spaces, you get bounds for the Sobolev constant as well. Such spaces include for example the Ricci limits (which were extensively studied by Cheeger and Colding).

This is true at least if you bound the curvature from below with some positive constant, see [Lott and Villani, Weak curvature conditions and Poincaré inequalities, J. Funct. Anal. 245 (2007), 311-333.] (Available also at http://math.univ-lyon1.fr/~villani/Cedrif/034.LV-Poincare.pdf )

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Thank you Dr. Rajala. One further question: do you know of an analogue of ricci bound from below in terms of bounding some integral of the quantity $-inf_{|x|=1} Ric(x,x)$? –  S.A.A Jun 27 '12 at 2:56
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