Related papers: Minimal quadrangulations of surfaces

We show that if G is a 4-critical graph embedded in a fixed surface $\Sigma$ so that every contractible cycle has length at least 5, then G can be expressed as $G=G'\cup G_1\cup G_2\cup ... \cup G_k$, where $|V(G')|$ and $k$ are bounded by…

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the…

In this paper we show that in addition to the known minimal surfaces which appear in the literature for computing the entanglement entropy there are other minimal surfaces with non-zero extrinsic curvature. We use the approach of…

Let $G$ be a simple finite graph and $G'$ be a subgraph of $G$. A $G'$-design $(X,\cal B)$ of order $n$ is said to be embedded into a $G$-design $(X\cup U,\cal C)$ of order $n+u$, if there is an injective function $f:\cal B\rightarrow \cal…

We prove that the least area of the non-contractible immersed spheres is no more than $4\pi$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ with nonzero…

For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic.…

We establish the lower bound of $4\pi(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\mathbb{R}^4$. This lower bound is…

Generalizing the Cauchy-Riemann equations, we construct the Osserman system of the first order for a pair $\left(f(x, y), g(x,y) \right)$ of two ${\mathbb{R}}$-valued functions on the domain $\Omega \subset {\mathbb{R}}^{2}$. The graph…

Minimal surfaces and Einstein manifolds are among the most natural structures in differential geometry. Whilst minimal surfaces are well understood, Einstein manifolds remain far less so. This exposition synthesises together a set of…

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4\pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not…

In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we…

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…

If a graph has $n\ge4k$ vertices and more than $n^2/4$ edges, then it contains a copy of $C_{2k+1}$. In 1992, Erd\H{o}s, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle is at least $2\lfloor…

We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2)…

Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T,…

For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of…

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,\tau)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,\tau)$ is such a…

Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold Z$ to be a…

Let $M$ be a 5 dimensional Riemannian manifold with $Sec_M\in[0,1]$, $\Sigma$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists $\varepsilon_0>0$, such that $\int_\Sigma (1+H^2)^2 \ge…