Planelike minimizers in periodic media

We show that given an elliptic integrand J in R-d that is periodic under in teger translations, and given any plane in Rd, there is at least one minimi zer of J that remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to R-d/Z(d), the minimizers we construct give rise to a lamination. One pa rticular case of these results is minimal surfaces for metrics invariant un der integer translations. The same results hold for other functionals that involve volume terms (smal l and average zero). In such a case the minimizers satisfy the prescribed m ean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that t heir fundamental group and universal cover satisfy some hypotheses. (C) 200 1 John Wiley & Sons, Inc.