  Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 2$.
This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy
inequality \[ {\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} +
B_{opt}\right) \] for all $u \in BV(M)$ with $\|u\|_{L^1(M)} = 1$ and existence
of extremal functions. In particular, we prove that this optimal inequality is
equivalent a optimal $L^1$-Sobolev inequality obtained by Druet [6].
