  We study the $S=1/2$ Heisenberg model on the triangular lattice with nearest
neighbor interaction $J_1$ and next nearest neighbor interaction $J_2$ with the
density matrix renormalization group. We are able to study long open cylinders
with widths up to 9 lattice spacings. At an intermediate $J_2$ region $0.06
\lesssim J_2/J_1 \lesssim 0.17$, we find evidence for a spin liquid (SL) state
with short range spin-spin, bond-bond and chiral correlation lengths, bordered
by a classical $120^\circ$ N\'eel ordered state at small $J_2$ and by a two
sub-lattice collinear magnetically ordered state at larger $J_2$. Focusing on
$J_2/J_1 = 0.1$, we find a number of signatures of a gapped SL phase: two
quasi-degenerate ground states on even cylinders, with an energy gap that
decreases exponentially with the cylinder width; a dimerization effect on odd
cylinders; and large spin triplet and singlet bulk gaps.
