ON THE GLOBAL STRUCTURE OF CRYSTALLINE SURFACES

We bound the number of plane segments in a crystalline minimal surface S in terms of its Euler characteristic, the number of line segments in its boundary, and a factor determined by the Wulff shape W of its surface energy function. A major technique in the proofs is to quantize the Gauss map of S based on the Gauss map of W. One thereby bounds the number of positive-curvature corners and the interior complexity of S.