论文标题
伯恩斯坦定理，用于尺寸四的两值最小图
A Bernstein theorem for two-valued minimal graphs in dimension four
论文作者
论文摘要
我们证明了四维欧几里得空间中的两值最小图的伯恩斯坦型定理$ \ mathbf {r}^4 $。这指出在整个$ \ mathbf {r}^3 $上定义的两个值函数，其图是最小的表面，必须是线性的。这是经典伯恩斯坦定理的两个值类似物，它断言在尺寸最高为$ n+1 \ leq 8 $中，整个单值最小图是线性的。与单价值理论的主要对比是在两个值函数的图中存在大量的奇异点。实际上，两个值的最小图既不是区域最小的，也不是椭圆形PDE的规律性理论，在这种情况下也可以直接可用。我们获得了两值最小图的排列锥的结构结果，在尺寸$ n+1 \ leq 7 $中有效，特别是证明了它们从$（N-2）$ - 可回座的套件中平稳地沉浸在包括其分支点的套件中。在第四维度中，我们走得更远，并使用组合参数完全对可能的排污锥进行了分类。我们表明它们必须是两种三维平面的结合：这是伯恩斯坦定理证明的关键。
We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must necessarily be linear. This is a two-valued analogue of the classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq 8$, an entire single-valued minimal graph is linear. The main contrast with the single-valued theory is the presence of a large set of singularities in the graphs of two-valued functions. Indeed two-valued minimal graphs are neither area-minimising, nor is the regularity theory of elliptic PDE directly available in this setting. We obtain structure results for the blowdown cones of two-valued minimal graphs, valid in dimension $n+1 \leq 7$, proving in particular that they are smoothly immersed away from an $(n-2)$-rectifiable set that includes its branch points. In dimension four we go further, and completely classify the possible blowdown cones using a combinatorial argument. We show that they must be a union of two three-dimensional planes: this is the key to the proof of the Bernstein theorem.
