Planelike minimizers in periodic media

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Abstract
We show that given an elliptic integrand π’₯ in ℝd that is periodic under integer translations, and given any plane in ℝd, there is at least one minimizer of π’₯ 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 ℝd/β„€d, the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations.The same results hold for other functionals that involve volume terms (small and average zero). In such a case the minimizers satisfy the prescribed mean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses. Β© 2001 John Wiley & Sons, Inc.

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