Nevanlinna Theory and Minimal Surfaces

Abstract

In 1915, S. Bernstein proved that there is no nonflat minimal surface in R 3 which is described as the graph of a C 2-function on the total plane R 2 ([9]). This result was improved by many researchers in the field of differential geometry. E. Heinz studied nonflat minimal surfaces in R 3 which are the graph of functions on discs Δ R := {(x, y); x 2 + y 2 < R 2} and showed that there exists a constant C > 0 such that |K(0)| ≤ C/R 2 for the Gaussian curvature K(0) at the origin ([43]). After some related results were given by E. Hopf ([45]), J. C. C. Nitsche ([52]) and so on, in 1961 R. Osserman proved that the Gauss map of a nonflat complete regular minimal surface in R 3 cannot omit a set of positive logarithmic capacity in the Riemann sphere ([56]) and, in 1981 F. Xavier showed that the Gauss map of such a surface can omit at most six values. Moreover, in 1988 the author gave the best possible version of this, which asserts that the number of exceptional values of the Gauss map of a complete nonflat regular minimal surface is at most four. Recently, several related results were given by X. Mo and R. Osserman ([49]), S. J. Kao ([48]) and M. Ru ([60]).KeywordsRiemann SurfaceMinimal SurfaceMeromorphic FunctionCompact Riemann SurfaceHolomorphic CurveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.