Analyticity of classical steady needle crystals

This paper is a continuation of our previous work “Rigorous results in selection of steady needle crystals, J. Differential Equations 197 (2004) 349–426”. It concerns analyticity of a classical steadily translating needle crystal. It is proved that any classical solution to the needle crystal problem with sufficiently small but nonzero surface tension, if its slope deviation is close to some Ivantsov zero-surface-tension solution and if its curvature satisfies some algebraic decay conditions at ∞, must belong to the analytic function space A0 defined in §1 and chosen in the previous study mentioned above. The analyticity result implies that there can be no classical steady needle crystal solution when anisotropy is zero.