Let $M$ be a compact manifold without boundary. Does the Nash inequality $$\lVert u \rVert^{1+\frac 2n}_{L^2} \leq C\lVert u \rVert^{\frac 2n}_{L^1} \lVert \nabla u \rVert_{L^2}$$ or something similar hold on $M$?
From what I read in the book by Hebey, "Poincare's inequality iff Nash's inequality", but that is old, maybe something better is known now. I can make $M$ as smooth as needed.
The way you asked the question, the answer is obviously negative, take any constant function different from 0. If M is not connected then even zero average is not enough to guarantee the inequality.