Related papers: Wild singularities of flat surfaces

We give the first examples of flows which exhibit robust singular attractors containing a wild hyperbolic set (in the sense of Newhouse). A hyperbolic set is said to be wild, if it has tangencies between its stable and unstable manifolds,…

We study the local geometry of the pullback of a variety via a finite holomorphic map. In particular, we are looking for properties of $V = F^{-1}(W)$ such that if $V$ has the property $A$, then $W$ must have the property $A$. We show that…

Let X be a complex algebraic variety, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. We show that the formal neighborhood of f in L(X) admits a decomposition into a…

We define local residues of holomorphic 1-forms on an isolated surface singularity that have isolated zeros and prove that a certain residue equals the index of the 1-forms.

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to…

Consider a polynomial F such that each variable appears in exactly one monomial. The hypersurface defined by the polynomial F is called a hypersurface with separable variables. A variety is called rigid if there are no nontrivial actions of…

Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that…

We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries…

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional type, show that they can generically have only finitely many singular points, and…

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…

We consider a mathematical model of synthetic aperture radar (SAR) with a known, possibly non-flat, topography. In this context we consider the problem of recovering the wavefront set of the ground reflectivity, given radar data measured…

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type…

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of…

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a…

Consider a complex one dimensional foliation on a complex surface near a singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an invariant real…

If $X$ is a (topological) space, the $n$th finite subset space of $X$, denoted by $X(n)$, consists of $n$-point subsets of $X$ (i.e., nonempty subsets of cardinality at most $n$) with the quotient topology induced by the unordering map…

We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial…

We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting…

The goal of this article is to prove that every surface with a regular point in the three-dimensional projective space of degree at least four, is of wild representation type under the condition that either $X$ is integral or…