Lipschitz normally embedded set and tangent cones at infinity

Lipschitz正规嵌入集的切锥是Lipschitz正规嵌入的。附录：Anne Pichon和Walter D. Neumann

Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. Appendix by Anne Pichon and Walter D. Neumann

证明了Lipschitz正规嵌入集的切锥是Lipschitz正规嵌入的。我们还扩展到真正的亚解析集的概念，减少切锥，我们表明，亚解析Lipschitz正常嵌入集有减少切锥。特别地，我们得到了Lipschitz正常嵌入复解析集具有约化切锥。

Testing Lipschitz non normally embedded complex spaces

We introduce a sectional criterion for testing if complex analytic germs $(X,0) \subset (\bC^n, 0)$ are Lipschitz non normally embedded.

关于拟反演

On Quasi-inversions

给定一个关于$0 \in D\,,$严格星形的有界域$D \subset {\mathbb R}^n$，我们定义了一个关于.边界$\partial D \,.$我们证明了拟反演是双Lipschitz w.r.t.弦度量当且仅当$\partial D$的每条“切线“都远离原点。当$\partial D$以适当的方式接近单位球时，双Lipschitz常数趋于$1,$.对于我们的结果的公式化，我们使用的概念的$α$ -正切条件，由于F。W. Gehring and J. Väisälä（Acta Math. 1965）.这个条件被证明是等价的双Lipschitz和拟共形扩展属性，我们称之为极性参数化的$\partial D$。此外，我们表明，极参数化，这是一个映射的单位球到$\partial D\,,$是bi-Lipschitz当且仅当$D$满足$α$ -正切条件。

On Lipschitz rigidity of complex analytic sets

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, a complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.

复集的重数和度作为双Lipschitz不变量

Multiplicity and degree as bi-Lipschitz invariants for complex sets

研究了复解析芽的重数和复仿射集的度在外双Lipschitz变换下的不变性（第一种情况是芽的外双Lipschitz同胚，第二种情况是芽在无穷远处的外双Lipschitz同胚）.我们证明了局部情况下的重数不变性等价于全局情况下的度不变性。我们证明了曲线和曲面的不变性。在这种方式中，我们证明了切锥的不变性和相对多重性在无穷远下的外双Lipschitz同胚在无穷远，和一个齐次表面芽的抽象拓扑决定其多重性。

Some classes of homeomorphisms that preserve multiplicity and tangent cones

In this paper we present some applications of A'Campo-Lê's Theorem and we study some relations between Zariski's Questions A and B. It is presented some classes of homeomorphisms that preserve multiplicity and tangent cones of complex analytic sets. Moreover, we present a class of homeomorphisms that has the multiplicity as an invariant when we consider right equivalence and this class contains many known classes of homeomorphisms that preserve tangent cones. In particular, we present some effective approaches to Zariski's Question A. We show a version of these results looking at infinity. Add…