Minimal
Surfaces
An important problem in minimal surface theory was to find the maximum number of points omitted by the Gauss map of a complete minimal surface. Following earlier work of Bernstein in non-linear partial differential equations and the important contributions of Osserman, Chern and Ahlfors, in [4] we proved that the Gauss map of a non-flat complete minimal surface (cms, for short) cannot omit seven points of the unit sphere. This result, along the lines of the Picard theorem in complex analysis, was met with great interest because, prior to it, the best known result, due to Osserman, even allowed for the omitted set to be uncountable. The optimal number of omitted points is four, as it was later proven by Fujimoto.
A natural question in connection with the Gauss map of minimal surfaces is the following: what meromorphic functions appear as the Gauss map of a cms? In [8] it is shown, using techniques from [4], that if a (perhaps surjective) meromorphic g is Lipschitzian from the Poincare disc into the Riemann sphere, with a Lipschitz constant smaller that 0.7, then g is not the Gauss map of a cms (a meromorphic function which omits three points is automatically Lipschitz in the above sense). In addition, it was shown that if the Lipschitz constant is less than 1 then g is not the Gauss map of a cms of bounded curvature.
In [2] we answer a question of Calabi and give an example of a cms that is non-flat and is contained in the region between two parallel planes. In a recent paper in Inventiones, Nadirashvili constructs a bounded immersed cms in 3-space, thus answering the second half of the Calabi question. His arguments are in the spirit of those of [2]. In [6] we use, for the first time in minimal surface theory, the Lusin area integral from classical harmonic analysis to prove the (sharp) result that the convex hull of a non-flat cms of bounded curvature is the entire space. Subsequently, Hoffman and Meeks established the same result when the cms is properly immersed.
An important open problem in this theory is the existence of a simply- connected embedded cms, other than the plane and the helicoid.In [13] we use function theory to study this problem. In our recent paper [17] we make significant progress on this question and obtain results that are geometrically very satisfying. Improving considerably on a previous result of Rodriguez and Rosenberg, we show that if a simply-connected embedded cms has bounded curvature it has to be one the two existing examples if, in addition, some plane cuts the surface transversally on a single (connected) curve. We also prove that if the curvature is strictly negative and bounded, then the surface has to be the helicoid if, in addition, there is a plane that intersects the surface on finitely many curves, the intersection being transversal except perhaps at finitely many points. These results are established using a powerful new geometric criterion for the existence of non-tangential limits for holomorphic functions defined in the unit disc. This latter result uses classical work of Fatou, Lusin and Privalov, among others. Meeks and Rosenberg have recently circulated a preprint with the solution of the helicoid problem when the surface is properly embedded. The properness assumption is unnatural in this context and one of our programs is to use the techniques from the classical theory of univalent functions to study embedded minimal discs. Initial progress has been made but more needs to be done.
Related Publications
[1] F. Xavier (with L. P. Jorge), On the existence of complete bounded minimal surfaces in $R^n$, Boletim da Sociedade Brasileira de Mathematica, 10, No. 2 (1979).
[2] F. Xavier (with L. P.Jorge), A complete minimal surface between two parallel planes, Annals of Mathematics, 112, 203-206, (1980).
[3] F. Xavier (with L. P.Jorge), An inequality between the exterior diameter and the mean curvature of bounded immersions, Mathematische Zeitschrift, 178, 77-82, (1981).
[4] F. Xavier, The Gauss map of complete minimal surface cannot omit 7 points of the sphere, Annals of Mathematics, 113 (1981) (erratum on 115), 211-214 .
[6] F. Xavier, Convex hulls of complete minimal surfaces, Math. Ann. 269 l79-l82, (1984 ).
[8] F. Xavier (with A. Weitsman), Some function theoretic properties of
the Gauss map for hyperbolic complete minimal surfaces,
[13] F. Xavier, On the structure of complete simply connected embedded minimal surfaces, Journal of Geometric Analysis, 3, No. 5, pp 513-527 (1993).
[17] F. Xavier, Embedded, simply connected, minimal surfaces with bounded curvature, GAFA, 11, 1344-1356 (2001).