Related papers: The embedded Nash problem in singular spaces: the …

We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.

We give simple criteria for the singularities appearing on surfaces codimension less than or equal to two. As applications, we give conditions for codimension two singularities that appear in ruled surfaces and center maps of surfaces in…

This paper shows the affirmative answer to the local Nash problem for a toric singularity and analytically pretoric singularity. As a corollary we obtain the affirmative answer to the local Nash problem for a quasi-ordinary singularity.

Tangent cones are preserved under ambient bilipschitz equivalence, but the behavior of the Nash cone is more delicate. This paper explores the behavior of the Nash cone and of exceptional rays under ambient bilipschitz equivalence for real…

We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.

Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given…

Let $(X,O)$ be a germ of a normal surface singularity, $\pi : \tilde X\longrightarrow X$ be the minimal resolution of singularities and let $A=(a_{i,j})$ be the $n\times n$ symmetrical intersection matrix of the exceptional set of $\tilde…

We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.

This study aims to investigate the functional properties of weak solution spaces and their compact embedding properties in relation to the Dirichlet problem associated with a specific class of degenerate elliptic equations. To expand the…

In this paper we introduce a maximal divisorial set in the arc space of a variety. The generalized Nash problem is reduced to a translation problem of the inclusion of two maximal divisorial sets. We study this problem and show a counter…

We propose a global minimal embedding of the Schwarzschild theory in a five-dimensional flat space by using two surfaces. Covariant field equations are deduced for the gravitational forces.

We present a quite simple analytical study on the appearance or absence of naked singularities in binary systems. As an example we consider the double Reissner-Nordstr\"om solution and fix the conditions it should satisfy in order to avoid…

The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ (S,O) of a singular surface as there are essential divisors over (S,O). It is known that this problem can be reduced to the study…

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with…

In this paper we give a positive answer to a question of Nash concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii.

Noether's theorem on the equivalence of symmetry and conservation laws has applications to geometric problems on symmetric spaces. We remind the reader of the theorem and give an application to a variational problem on hyperbolic surfaces.

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.

Trapped surfaces are studied as inner boundary for the Einstein vacuum constraint equations. The trapped surface condition can be written as a non linear boundary condition for these equations. Under appropriate assumptions, we prove…

This paper seeks to prove the bijectivity of the "Nash mapping" from the set of irreducible components of the scheme parametrizing analytic arcs on an algebraic surface $X$ whose origin is a singular point, into the set of irreducible…

We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend…