极小子流形的一个二分性
A dichotomy for minimal submanifolds
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中文总结 AI 辅助
本文综述最新成果，提出极小子流形的二分性，通过体积加倍定理等机制推导其受限性带来的几何与函数论限制，将$\bf{R}^3$嵌入极小圆盘理论弱推广至所有维数。
中文摘要 AI 辅助
我们综述一系列最新研究成果，表明极小子流形满足一种二分性：要么它像空间填充曲线一样向外延展以填满空间，要么它是受限的，而受限性会带来定量限制。在受限侧，这些限制兼具几何与函数论性质：欧氏体积增长、密度的最优收敛速率、$\bf{R}^4$中稳定曲面的复曲线刚性，以及迫使极小圆盘上缓慢增长的调和函数为常数的刘维尔定理。一个典型限制是欧氏体积增长，在这些结果中它由几何性质而非假设所强制。其机制是一个体积加倍定理，该定理将几何受限性转化为定量刚性：在给定尺度下被困于薄平板中的稳定积分 varifold，其体积加倍的倍数不超过一个普适常数。我们阐释该原理及其背后的高度过剩界，如何在所有维数与余维数下，为高度次线性增长的子流形导出欧氏体积增长、平板中的最优密度速率、上述复曲线与刘维尔刚性、圆盘的高余维伯恩斯坦定理、单侧体积界，以及推广了Moser、Bombieri-De Giorgi-Miranda、Caffarelli-Nirenberg-Spruck和Ecker-Huisken的所有维数下的最优稳定伯恩斯坦定理。我们还阐释了整个图景如何从$\bf{R}^3$中嵌入极小圆盘的结构理论中产生，该理论建立在单侧曲率估计之上，并在半空间定理中得到全局体现，以及它如何作为该理论的弱类似物推广到所有维数。
英文摘要
We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in $\bf{R}^4$, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We also explain how this entire picture emerges from the structure theory of embedded minimal disks in $\bf{R}^3$, built on the one-sided curvature estimate and reflected globally in the half-space theorem, and how it extends as a weak analogue of that theory to all dimensions.
