A Complete Minimal Surface in R 3 Between two Parallel Planes

Abstract

E. Calabi has asked if it is possible to have a complete minimal surface in RX entirely contained in a half-space. We answer his question in the affirmative. In fact, we prove considerably more, namely, we show how to produce complete minimal surfaces contained in slabs of R3. The proof was motivated by Remmert's ingenious idea of using Runge's theorem to exhibit proper analytic embeddings of the unit disc D c( C in C3, as explained in [11,