We solve the general-relativistic steady-state eigenvalue problem of neutrino-driven protoneutron star winds , which immediately follow core-collapse supernova explosions . We provide velocity , density , temperature , and composition profiles and explore the systematics and structures generic to such a wind for a variety of protoneutron star characteristics . Furthermore , we derive the entropy , dynamical timescale , and neutron-to-seed ratio in the general relativistic framework essential in assessing this site as a candidate for r -process nucleosynthesis . Generally , we find that for a given mass outflow rate ( \dot { M } ) , the dynamical timescale of the wind is significantly shorter than previously thought . We argue against the existence or viability of a high entropy ( \gtrsim 300 per k _ { B } per baryon ) , long dynamical timescale r -process epoch . In support of this conclusion , we model the protoneutron star cooling phase , calculate nucleosynthetic yields in our steady-state profiles , and estimate the integrated mass loss . We find that transonic winds enter a high entropy phase only with very low \dot { M } ( \lesssim 1 \times 10 ^ { -9 } M _ { \odot } s ^ { -1 } ) and extremely long dynamical timescale ( \tau _ { \rho } \gtrsim 0.5 seconds ) . Our results support the possible existence of an early r -process epoch at modest entropy ( \sim 150 ) and very short dynamical timescale , consistent in our calculations with a very massive or very compact protoneutron star that contracts rapidly after the preceding supernova . We explore possible modifications to our models , which might yield significant r -process nucleosynthesis generically . Finally , we speculate on the effect of fallback and shocks on both the wind physics and nucleosynthesis . We find that a termination or reverse shock in the wind , but exterior to the wind sonic point , may have important nucleosynthetic consequences . The potential for the r -process in protoneutron star winds remains an open question .