The strong halfspace theorem for minimal surfaces
D. Hoffman and W.H. Meeks, III Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA In [6] Jorge and Xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in •3. Recently Rosenberg and Toubiana have found complete minimally immersed annuli that are proper in an open slab 1-12]. On the basis of these and other results, the question has arisen as to whether or not there could exist a properly immersed minimal surface in ~3 that was contained between two parallel planes. The answer is negative, an immediate consequence of the following theorem.
We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of ...