Related papers: Local Geometry of Singular Real Analytic Surfaces

We apply the local removable singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two…

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…

We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…

We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity…

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the…

This note is about promoting singularity subtraction as a helpful tool in the discretization of singular integral operators on curved surfaces. Singular and nearly singular kernels are expanded in series whose terms are integrated on…

In this article we extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over such a manifold admits a parallel symmetric 2-tensor then it is incomplete and has non zero constant…

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…

Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric $\omega$ with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a…

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small…

This paper locally classifies finite-dimensional Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$ relative to an arbitrary pseudo-Riemannian metric. Several results about their geometric properties are detailed, e.g.…

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in…

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class…

Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials…

The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation. In all…

We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate…

Two subanalytic subsets of $ \mathbb R^n$ are called $s$-equivalent at a common point $P$ if the Hausdorff distance between their intersections with the sphere centered at $P$ of radius $r$ vanishes to order $>s$ as $r$ tends to $0$. In…

In this short note we outline a simple probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, which adapts to this setting an argument due to Hsu \cite{Hs1,Hs2} in the closed case. The new technical…

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite…