Zoom URL: https://oist.zoom.us/j/95676276965?pwd=QTN2QjBVRWwwL01Dbm1ndFBQa2FTQT09

Bernstein's theorem for minimal surfaces and its generalization

Prof. Min Ru

Department of Mathmatics, University of Houston

Abstract: A surface in R^3 is called minimal if the mean curvature of the surface is zero at every point. In 1916, Bernstein proved the following result: If a surface of graph z=f(x,y) defined on the whole xy-plane is minimal, then it must be planar, meaning that f(x,y) is a linear function. Later, Nirenberg took the view about the range of its Gauss map (i.e., the field of unit normals of the surface z=f(x,y)is confined to a hemisphere) and raised the following conjecture for general minimal surfaces immersed in R^3: A complete simply-connected minimal surface in R^3 must be planar if the Gauss map omits a neighborhood of some direction. Osserman in 1959–1965 used techniques from complex analysis to show that Nirenberg's conjecture is true. Inspired by Osserman's proof and by comparison to the classical little Picard's theorem in complex analysis, Xavier showed that the Gauss map of a non-planar complete minimal surface in R^3 cannot omit 7 directions. Finally, Fujimoto in 1988 refined Xavier's result from 7 to 5, which is sharp: the Gauss map of the classical Scherk surface omits four points on the unit hemisphere. Complex analysis, especially Nevanlinna theory (i.e., the theory of holomorphic curves), is central to the proofs of Xavier and Fujimoto. Indeed, in the mid 1960s, Chern and Osserman initiated a program of using Nevanlinna theory to study the value distribution properties of the (generalized) Gauss maps of minimal surfaces immersed in R^m. Since then, there have been many important developments. In this talk, I will survey recent progress in this direction.