论文标题
在复杂分析集的无穷大的Lipschitz几何形状上
On Lipschitz Geometry at infinity of complex analytic sets
论文作者
论文摘要
在本文中，我们研究了复杂分析集的无穷大的Lipschitz几何形状，并获得了分析集和伯恩斯坦问题的代数性的结果。 Moser的Bernstein Theorem说，整个Lipschitz函数的最小超出表面必须是超平面。 H. B. Lawson，Jr。和R. Osserman提出了例子，表明任意编成的类似结果是不正确的。在本文中，我们证明了Moser的Bernstein Theorem的复杂非参数版本。更确切地说，我们证明，在$ \ mathbb {c}^n $中设置的任何整个复杂分析都设置为Infinity时Lipschitz常规的affine线性子空间，$ \ mathbb {c}^n $。特别是，一个是整个Lipschitz函数图的复杂分析集必须是仿射线性子空间。该结果是由于以下代数集的以下表征而产生的，在这里也证明了这一点：如果$ x $和$ y $是整个复杂的分析集，则是无限的bi-lipschitz同型同构，那么$ x $是一个复杂的代数集，并且仅当$ y $是一个复杂的代数集合时，也是一个复杂的代数。因此，整个复合物分析集是一个复杂的代数集，并且仅当它是Infinity的Bi-lipschitz同型对复杂代数集的同构时。在此处证明的结果中，对单数集，维度和编纂不需要限制。
In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity must be an affine linear subspace of $\mathbb{C}^n$. In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if $X$ and $Y$ are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then $X$ is a complex algebraic set if and only if $Y$ is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.
