  We use the (M,n) phase diagram recently introduced in hep-th/0309116 to
investigate the phase structure of black holes and strings on cylinders. We
first prove that any static neutral black object on a cylinder can be put into
an ansatz for the metric originally proposed in hep-th/0204047, generalizing a
result of Wiseman. Using the ansatz, we then show that all branches of
solutions obey the first law of thermodynamics and that any solution has an
infinite number of copies. The consequences of these two results are analyzed.
Based on the new insights and the known branches of solutions, we finally
present an extensive discussion of the possible scenarios for the
Gregory-Laflamme instability and the black hole/string transition.
