The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds
Zhenhua Liu
TL;DR
This work establishes a geometric-analytic Hasse principle for area-minimizing currents, showing that integral-homology minimizers can be reconstructed from real and mod $n$ data across all $n\,\ge 2$. The authors develop a comprehensive framework of coefficient norms, Federer's representative modulo $n$, calibration theory, and metric-monotonicity to compare minimizers across coefficients, proving that large-$n$ reductions yield bijections with integral minimizers under suitable divisibility, while real and mod-$n$ reductions can also fail to preserve minimizers in other settings. The results imply surprising regularity and calibration phenomena, including asymptotic smoothness for mod $n$ minimizers and the non-generic calibration of $ obreak\mathbb{Z}$-minimizers, and reveal that products of area-minimizing submanifolds are not generically area-minimizing. The paper further proposes broad conjectures and analogies across Plateau problems, Künneth-type questions, and Ostrowski-type norm theories, suggesting a rich landscape of local-to-global principles for geometric variational problems. Overall, the work offers a unifying principle linking integral, real, and modular coefficient theories in geometric measure theory with potential implications for broader variational frameworks.
Abstract
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 the Hasse principle holds for area-minimizing submanifolds: information about area-minimizing submanifolds in integral homology can be fully recovered from those in real homology and mod n homology for all $n\in \mathbb{Z}_{\ge 2}$. As a consequence we derive several surprising conclusions, including: area-minimizing submanifolds in mod n homology are asymptotically much smoother than expected, area-minimizing submanifolds are not generically calibrated, and products of area-minimizing submanifolds are not generically area-minimizing. We conjecture that the Hasse principle holds for all geometric variational problems that can be formulated on chain space over different coeffiicients, e.g., Almgren-Pitts min-max, mean curvature flow, Song's spherical Plateau problem, minimizers of elliptic and other general functionals, etc.