Related papers: Surfaces given with the Monge patch in E^4

The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on…

We survey what is known about minimal surfaces in $\bold R^3 $ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early…

For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the…

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially…

Chapters : Old and new inequalities; Surfaces with $\chi=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF…

We prove results showing that the existence of essential maps of surfaces in a manifold M' obtained from a 3-manifold M by Dehn filling implies the existence of essential maps of surfaces in M.

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of…

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…

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of…

In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of…

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant…

In this paper we consider the complete biconservative surfaces in Euclidean space $\mathbb{R}^3$ and in the unit Euclidean sphere $\mathbb{S}^3$. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the…

We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $\Sigma$ is a complete, properly…

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean…

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type $a H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the Gauss…

We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean $n$-space ($n$=3 or 4), improper affine…

A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in…

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our…