The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds

The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that an analogous Hasse principle holds for area-minimizing submanifolds: information about area-minimizing submanifolds in integral homology can be recovered from those in real homology and mod $n$ homology for all $n\in {\mathbb{Z}}_{\ge 2}$. As a consequence we answer several questions of Almgren, Morgan and White and prove: area-minimizing submanifolds are not generically calibrated, products of area-minimizing submanifolds are not generically area-minimizing, and classification of area-minimizing pairs of planes mod $n$ for $n\ge 4$.