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}$.