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Consider a minimal surface x=(xlt ••• , xm): M-+R m in R. In the previous papers [9], [10], [11] and [12], the author gave some value-distribution-theoretic properties of the Gauss map of M in the case where M is complete (cf., [13]). On the other hand, under some assumptions, E. F. Beckenbach and his collaborators showed that the map x itself has many properties which are similar to the results in Nevanlinna theory for meromorphic functions on C in their papers [4], [3], [2] and [6]. They developed their theory for 'meromorphic minimal surfaces'. Roughly speaking, these are minimal surfaces in R with at worst pole-like singularities which is conformally isomorphic with the complex plane. The purpose of this paper is to extend some of their results to the case where M is conformally isomorphic with a Riemann surface of parabolic type. For brevity, we restrict ourselves to the case of regular minimal surfaces though our arguments are also available for minimal surfaces with pole-like singularities. By definition, a Riemann surface M is of parabolic type if there is a proper map r: Af-»[0, +00) of class C°° such that dd\ogτ=Q and ddτ^Q on M—MS for some s>0, where Ms :— {αeM; τ(α)<s}. We define the hyperspherical function by

Nevanlinna theory for minimal surfaces of parabolic type