Related papers: On Lipschitz Normally Embedded singularities
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…
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…
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…
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…
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…
We study the ambient Lipschitz geometry of semialgebraic surfaces. It was discovered in \cite{BBG} that ambient Lipschitz Geometry is different from the outer Lipschtz geometry. We show that two surface germs in $\mathbb{R}^3$, Lipschitz…
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…
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…
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…
We study metric properties of manifolds with conic singularities and present a natural interplay between metrically conic and metrically asymptotically conic behaviour. As a consequence, we prove that a singular sub-manifold is Lipschitz…
The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically…
The aim of this paper to introduce the reader to a recent point of view on the Lipschitz classifications of complex singularities. It presents the complete classification of Lipschitz geometry of complex plane curves singularities and in…
We introduce a sectional criterion for testing if complex analytic germs $(X,0) \subset (\bC^n, 0)$ are Lipschitz non normally embedded.
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…
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…
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 H\"older triangle is either Lipschitz normally…
In this paper, we prove that two normal complex surface germs that are inner bilipschitz--but not necessarily orientation-preserving--homeomorphic, have in fact the same oriented topological type and the same minimal plumbing graph. Along…
Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an…
Let $(X, 0)$ be a normal complex surface germ embedded in $(\mathbb{C}^n, 0)$, and denote by $\mathfrak{m}$ the maximal ideal of the local ring $\mathcal{O}_{X,0}$. In this paper, we associate to each $\mathfrak{m}$-primary ideal $I$ of…
Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of…