标题： 几何变分问题的哈塞原理：通过面积最小化子流形进行说明 显示英文标题

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

摘要： 数论中的哈塞原理指出，关于不定方程的整数解的信息可以从实数解和模素数幂的解中拼凑出来。 我们证明哈塞原理适用于面积最小化子流形：关于积分同调中的面积最小化子流形的信息可以完全从实同调和模$n$同调中的那些信息中恢复，对于所有$n\in \mathbb{Z}_{\ge 2}.$。作为推论，我们得出几个令人惊讶的结论，包括：模$n$同调中的面积最小化子流形在渐近意义上比预期的要平滑得多，且面积最小化子流形并非通常情况下由校准给出。 我们猜想哈塞原理适用于所有可以在不同系数上的链空间上表述的几何变分问题，例如，Almgren-Pitts 极小极大问题、平均曲率流、Song 的球面 Plateau 问题、椭圆和其他一般泛函的极小值问题等。 显示英文摘要

摘要： 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 and area-minimizing submanifolds are not generically calibrated. 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.