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

TL;DR

This paper demonstrates that the Hasse principle applies to area-minimizing submanifolds, allowing their properties to be deduced from solutions over various coefficient systems, revealing unexpected regularity and non-minimality phenomena.

Contribution

It establishes the Hasse principle for geometric variational problems, connecting integral, real, and modular homology solutions, and suggests broad applicability to other variational problems.

Findings

Area-minimizing submanifolds in mod n homology are smoother than expected.

Such submanifolds are not generically calibrated.

Products of area-minimizing submanifolds are not necessarily area-minimizing.

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.