Tien-Son Pham, Nguyen Thao Nguyen Bui
We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the leading coefficients of the asymptotic expansions of $f$ along the connected components of the tangency variety of $f.$
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Consider a Henselian rank one valued field $K$ of equicharacteristic zero with the three-sorted language $\mathcal{L}$ of Denef--Pas. Let $f: A \to K$ be a continuous $\mathcal{L}$-definable (with parameters) function on a closed bounded subset $A \subset K^{n}$. The main purpose is to prove that then $f$ is Hölder continuous with some exponent $s\geq 0$ and constant $c \geq 0$, a fortiori, $f$ is uniformly continuous. Further, if $f$ is locally Lipschitz continuous with a constant $c$, then $f$ is (globally) Lipschitz continuous with possibly some larger constant $d$. Also stated are some pro…
We prove two principal kinematic formulas for germs of closed definable sets in $\mathbb{R}^n$, that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidian motions, but on the manifold $SO(n) \times S^{n-1}$.
A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^n\to \mathbb{R}^\ell$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the $p$-adic context that $\widetilde f$ can be taken definable when $f$ is definable, where definable mea…
Local quasi-isometries and tangent cones of definable germs
本文通过引入度量芽的局部拟等距概念，证明可定义芽的拟等距等价与切锥的双Lipschitz同胚等价，并由此重新证明Sampaio切锥定理及Fernandes-Sampaio关于Lipschitz正规嵌入芽的切锥保持性质。
Extension of Lipschitz maps definable in Hensel minimal structures
本文在Hensel极小字段中研究Lipschitz映射的扩展，提出可定义的开放单元包概念，结合risometry，通过双归纳法解决非局部紧非阿基米德情形下的Kirszbraun定理推广问题。
In this paper we study Lipschitz contact equivalence of continuous function germs in the plane definable in a polynomially bounded o-minimal structure, such as semialgebraic and subanalytic functions. We partition the germ of the plane at the origin into zones where the function has explicit asymptotic behavior. Such a partition is called a pizza. We show that each function germ admits a minimal pizza, unique up to combinatorial equivalence. We show then that two definable continuous function germs are definably Lipschitz contact equivalent if and only if their corresponding minimal pizzas are…
It is known that the bi-Lipschitz right classification of function germs admit moduli. In this article we introduce a notion called the Lipschitz simple function germs and present a full classification in the complex case. A surprising consequence of our result is that a function germ is Lipschitz modal if and only if it deforms to the smooth unimodal family of singularities called $J_{10}$ in Arnold's list.
The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.
On the density at infinity of definable functions
本文证明了可定义函数纤维在无穷远处的密度在渐近临界值集合外是局部利普希茨连续的。
A bi-Lipschitz invariant for analytic function germs
本文为双变量解析函数芽提出新的双Lipschitz不变量，改进了Henry-Parusinski不变量。
It is known by a result of Mendes and Sampaio that the Lipschitz normal embedding of a subanalytic germ is fully characterized by the Lipschitz normal embedding of its link. In this note, we show that the result still holds for definable germs in any o-minimal structure on $(\mathbb{R}, + , .)$. We also give an example showing that for homomorphisms between MD-homologies induced by the identity map, being isomorphic is not enough to ensure that the given germ is Lipschitz normally embedded. This is a negative answer to the question asked by Bobadilla et al. in their paper about Moderately Disc…
One point compactification and Lipschitz normally embedded definable subsets
本文证明了o-minimal结构中可定义闭子集的Lipschitz法向嵌入性等价于其单点紧化在单位球面上的相应性质，并由此得出欧氏空间中任意闭连通无界可定义子集可定义地内部bi-Lipschitz同胚于Lipschitz法向嵌入的可定义集。
These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which is locally definably bi-Lipschitz trivial along the strata. This result is obtained as a byproduct of two foregoing results of the author. The first one asserts that, given a family definable in an o-minimal structure, there is a regular vector, up to a definable family of bi-Lipschitz homeomorphisms. The second one is a bi-Lipschitz version of the famous H…
We prove that local Lipschitz Killing curvatures of definable sets in a polynomially bounded o-minimal structure are continuous along strata of Whitney stratifications and locally Lipschitz if the stratifications are (w)- regular.
In this paper, we establish the following criterion for divisibility in the local ring of those quasianalytic function germs at zero which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two such function germs is that of their Taylor series at zero in the formal power series ring.
In this short note, we consider the problem of bi-Lipschitz contact equivalence of complex analytic function-germs of two variables. It is inquiring about the infinitesimal sizes of such function-germs, up to bi-Lipschitz changes of coordinates. We show that this problem is equivalent to the problem of the right topological classification.
We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.
Classification of Lipschitz unimodal function germs
本文为孤立奇点引入Lipschitz模概念，完整分类了余秩2且4-jet非零的Lipschitz单模奇点，并证明6-jet消失时模至少为2，给出上界。
Inner Lipschitz Geometry of Complex Surface Germs with Non-Isolated Singularities: A Complete Classification
本文通过归一化映射坐标函数的数值不变量和适当好解消的组合学，给出了非孤立奇点复曲面芽内部Lipschitz几何的完全不变量，将Birbrair-Neumann-Pichon的结果推广到非孤立情形。
Definable Lipschitz selections for affine-set valued maps
本文研究了o-极小结构中可定义仿射集值映射的Lipschitz选择问题，证明了若此类映射存在Lipschitz选择，则必存在可定义的Lipschitz选择，从而解决了可定义的Brenner-Epstein-Hochster-Kollár问题及相关Whitney延拓猜想。
Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if…
In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of the well-known Fukuda-Fukuda theorem is provided.
We present here basic results in Lipschitz Geometry of semialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer metric remains an open problem. We review recent results related to the outer and ambient bi-Lipschitz classification of surface germs. In particular, we explain why the outer Lipschitz classification is much harder than the inner classification, and why the ambient Lipschitz Geometry of surface germs is very different from their outer Lipschitz Geometry. I…
In this survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and motivic type invariants are encompassed under this point of view. We then explain how all these invariants are related to e…
Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y)_{y\in\mathbb{R}^s}$ be a definable family of closed subsets of $\mathbb{R}^n$ whose total space $W = \cup_y W_y\times y$ is a closed connected $C^2$ definable sub-manifold of $\mathbb{R}^n\times\mathbb{R}^s$. Let $φ:W \to\mathbb{R}^s$ be the restriction of the projection to the second factor. After defining $K(φ)$, the set of generalized critical values of $φ$, showing that they are closed and definable of positive codimension in $\mathbb{R}^s$, contain the bifurcation values of $φ$ and are stable under generic …
Ambient Lipschitz geometry of normally embedded surface germs
本文研究半代数曲面的环境利普希茨几何，证明ℝ³中具孤立奇点的利普希茨正规嵌入曲面芽的环境双利普希茨等价性等价于外部双利普希茨等价性与环境拓扑等价性的结合。
We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs $(X_1,0)$ and $(X_2,0)$ in $(\mathbb C^3,0)$ having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones…
Let A subset R^n be a set-germ at 0 in R^n such that 0 is in the closure of A. Let D(A) denote the set of all directions of A at 0 in R^n. Let A, B subsets R^n be subanalytic set-germs at 0 in R^n such that 0 belongs to their closure. We study the problem of whether the dimension of the common direction set, called their kissing dimension, is preserved by a bi-Lipschitz homeomorphism. We show that in general it is not preserved. We prove that the kissing dimension is preserved if the images of the subanalytic sets under consideration are also subanalytic. In particular, if two subanalytic set-…
The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove that the model determinantal singularity, that is the space of $m\times n$ matrices of rank less than a given number, is Lipschitz normally embedded. We will also discuss some of the difficulties extending this result to the case of general determinantal singularities.
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。
A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that: the link at the origin of $X_{K}$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. We s…
bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs
本文证明了双变量复拟齐次多项式函数芽的bi-Lipschitz等价与解析等价等价，从而完成了该类的分类。
The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space $m \times n$ matrices, symmetric matrices and skew-symmetric matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant $0$. We also make a short discussion ab…
Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), i.e., to have bilipschitz equivalent outer and inner metrics. In a partner paper [15] we apply it to prove that rational surface singularities are LNE if and only if they are mini…
Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.
Multi-$\mathcal{K}$-Lipschitz equivalence in dimension two
本文通过引入Multipizza划分将平面多重芽转化为非阿基米德空间，证明了其在多重$\mathcal{K}$-Lipschitz等价下的存在性、唯一性与完全不变性。
We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.
We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\ne 0$.
We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topol…
Bi-Lipschitz triviality of function-germs on singular varieties
本文研究奇异簇上解析函数芽形变的双李普希茨平凡性，引入强有理双李普希茨平凡族概念并给出无穷小判别准则，进而得到齐次与加权齐次情形下的平凡性和刚性结果。
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is a complete rank one valued fields with the Tate algebra of strictly convergent power series. The algebraic case was treated in our previous papers. Here we are going to carry over the research to the general analytic settings. Also considered are the local rings of analytic function germs induced by a given separated Weierstrass system, which turn out to be…
On the Lipschitz saturation of toric singularities
本文研究环面奇点芽的Lipschitz饱和，证明具有光滑正规化的环面奇点芽的Lipschitz饱和仍为环面奇点，并展示了如何从定义半群计算特定环面奇点族的Lipschitz饱和。
In this paper we investigate the relation betwen the Nash modification and the Bi-Lipschtiz equivalent germs in the cases of two germs and for a family of hypersurfaces with isolated singularities.
We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $στ$-pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.
On the minimality of pancake decomposition of surface germs
本文针对捕捉外部Lipschitz现象的异常曲面（蛇与环蛇），提出获取最小煎饼分解的算法，并证明弱外部Lipschitz等价曲面的贪心煎饼分解具有弱等价性与典范性。
Let $P^{k}(n,p) $ be the set of all real polynomial map germs $f = ( f_1 , ..., f_p ) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p},0)$ with degree of $f_1 , ...,f_p$ less than or equal to $ k \in \mathbb{N}$. The main result of this paper shows that the set of equivalence classes of $ P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz equivalence, is finite.
Two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent
Lipschitz geometry of complex surface germs via inner rates of primary ideals
本文通过为准素理想关联连续函数，证明了该函数由复曲面芽的外部Lipschitz几何决定，并构造了正规化同构但Lipschitz类型不同的孤立奇点无穷族。
On Strong Bi-Lipschitz Triviality of Deformations
本文通过构造反例，证明Thom-Levine判据的逆命题在双Lipschitz范畴不成立，即存在双Lipschitz平凡但非强双Lipschitz平凡的形变。
We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.
For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence in terms of the real tree model. We also give several examples of bi-Lipschitz …
Local vs. global Lipschitz geometry
本文研究 o-minimal 结构中可定义集合链的内蕴距离等价性，并据此建立奇点局部与全局 Lipschitz 几何之间的联系，刻画外部 lipeomorphism、球极投影修改与反演之间的等价关系。
Real and complex integral closure, Lipschitz equisingularity and applications on square matrices
本文改进矩阵芽 Lipschitz 平凡性的已有结果，将积分闭包结论推广到实背景，并应用于 Bruce 和 Tari 分类的方阵奇异性研究。
We relate the Lipschitz-Killing measures of a definable set $X \subset \mathbb{R}^n$ in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of $\mathbb{R}^n$, such results were established by Langevin and Shifrin.Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.
Lipschitz geometry of the image of finite mappings
本文研究复解析超曲面参数化芽的LNE性质，证明了重数等于一般次数的有限解析映射芽的像在原点为LNE等价于其为光滑芽，还给出了余秩1映射及单射映射芽的相关LNE等价条件。
We introduce a sectional criterion for testing if complex analytic germs $(X,0) \subset (\bC^n, 0)$ are Lipschitz non normally embedded.
Bi-H"older invariants in o-minimal structures
研究o-最小结构中双Holder不变量的性质，证明双α-Holder等价的germ具有相同的LNE性质、切锥维度及同调群，从而推导出复解析germ的光滑性定理。
Bi-Lipschitz invariance of the transverse polynomial along polar arcs
本研究证明在双Lipschitz右等价下，约化全纯平面函数芽沿极弧的横截多项式在特定尺度范围内不变，并恢复增广牛顿多边形的双Lipschitz不变性，且横截多项式提供额外新信息。
We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.
Semialgebraic Lipschitz equivalence polynomial functions
本文通过将两个变量实拟齐次多项式的半代数双Lipschitz等价分类归结为高度函数的Lipschitz分类，并借助β-变换理论证明该类等价也容许连续模。
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