  We experimentally demonstrate that spin dynamics and the phase diagram of
spinor condensates can be conveniently tuned by a two-dimensional optical
lattice. Spin population oscillations and a lattice-tuned separatrix in phase
space are observed in every lattice where a substantial superfluid fraction
exists. In a sufficiently deep lattice, we observe a phase transition from a
longitudinal polar phase to a broken-axisymmetry phase in steady states of
lattice-confined spinor condensates. The steady states are found to depend
sigmoidally on the lattice depth and exponentially on the magnetic field. We
also introduce a phenomenological model that semi-quantitatively describes our
data without adjustable parameters.
