The kissing dimension of subanalytic sets is preserved by a bi-Lipschitz homeomorphism

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e any complex analytic set which is locally bi-lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu's result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.

模2重数不是双李普希茨不变量

The multiplicity mod 2 is not a bi-lipschitz invariant

该研究证明亚解析双李普希茨同胚一般不保持实解析集的重数，说明模2重数并非双李普希茨不变量。

几何方向丛的 Lipschitz 类型

The Lipschitz type of the Geometric Directional Bundle

本文研究 R^n 子集的几何方向丛在 bi-Lipschitz 同胚下的类型保持条件，并证明亚解析集的 bi-Lipschitz 像一般具有 SSP。

Directional properties of sets definable in o-minimal structures

In a former paper the first and third authors introduced the notion of direction set for a subset of R^n, and showed that the dimension of the common direction set of two subanalytic subsets, called directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem and discuss in detail directional properties in the case of an Archimedean real closed field, and…

On the Fukui-Kurdyka-Paunescu Conjecture

In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (resp. degree) of real and complex analytic (resp. algebraic) sets. For instance, still in the real case, we prove a global version of Fukui-Kurdyka-Paunescu's Conjecture. In the complex case, one of the results that we prove is the following: If $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ are germs of analytic sets and $h\c…

关于Lipschitz正规嵌入奇点

On Lipschitz Normally Embedded singularities

本文综述了Lipschitz正规嵌入奇点的一般理论，并重点介绍了复曲面情形下的判据、例子和性质，最后提出若干开放问题。