Alexandre Fernandes

On local diffeomorphisms ofR n that are injective

There are obtained conditions under which maps fromRn to itself are globally injective. In particular there are proved some partial results related to the Weak Markus-Yamabe Conjecture which states that if a vector field X:Rn →Rn has the property that, for allp ∈Rn, all the eigenvalues ofD X (p) have negative real part, thenX has at most one singularity.

BI-LIPSCHITZ GEOMETRY OF WEIGHTED HOMOGENEOUS SURFACE SINGULARITIES

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.

Bi-Lipschitz geometry of complex surface singularities

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface with particular emphasis on when it is metrically conical.

-bi-Lipschitz equivalence of real function-germs

In this paper we prove that the set of equivalence classes of germs of real polynomials of degree less than or equal to k, with respect to κ-bi-Lipschitz equivalence, is finite.

Separating sets, metric tangent cone and applications for complex algebraic germs

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.

Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariskis multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in

Real and Complex Singularities: On normal embedding of complex algebraic surfaces

\mathbb{C}^3

Topological K -equivalence of analytic function-germs

is a bi-Lipschitz invariant.

Multiplicity and degree as bi-Lipschitz invariants for complex sets: MULTIPLICITY AND DEGREE AS BI-LIPSCHITZ INVARIANTS

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.

Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane

We study the topological K-equivalence of function-germs (ℝn, 0) → (ℝ, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs.