Related papers: Tubes of finite Chen-type
This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…
We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the…
We prove the unpolarized Shafarevich conjecture for K3 surfaces: the set of isomorphism classes of K3 surfaces over a fixed number field with good reduction away from a fixed and finite set of places is finite. Our proof is based on the…
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key is to understand the structure of an embedded minimal disk in a ball in…
We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is…
We prove that a surface in Euclidean $3$-space has Maslovian normal bundle if and only if it is a part of a round sphere, a circular cylinder, or a circular cone.
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable…
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…
This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to…
Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the…
We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form…
For a K3 surface S, consider the subring of CH(S^n) generated by divisor and diagonal classes (with Q-coefficients). Voisin conjectures that the restriction of the cycle class map to this ring is injective. We prove that Voisin's conjecture…
Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the N\'eron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric…
We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a…
We introduce and study tame homeomorphisms of surfaces of infinite type. These are maps for which curves under iterations do not accumulate onto geodesic laminations with non-proper leaves, but rather just a union of possibly intersecting…
Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with…
This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also…
Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…
We define a new kind of helicoidal surface of value m. A rotational surface which is isometric to the helicoidal surface of value m is revealed. In addition, we calculate some differential geometric properties of the helicoidal surface of…
A classical question in geometry is whether surfaces with given geometric features can be realized as embedded surfaces in Euclidean space. In this paper, we construct an immersed, but not embedded, infinite $\{3,7\}$-surface in…