Related papers: Surfaces containing two isotropic circles through …

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n \ge 2 which admit a dominating family V of rational curves of H-degree 3, such that two general points of X may be joined by a curve parametrized by…

We consider triangulations of closed $2$-dimensional (not necessarily orientable) surfaces. Any minimal set of zigzags that double covers the set of edges provides a $z$-orientation of the triangulation. We introduce Markov chains of…

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal…

We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a…

A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form…

The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it…

We propose a unified method to visualize curvature on planar curves and surfaces of revolution using the tangential angle parameter. For plane curves, placing markers at equal increments of the tangential angle reveals local bending…

In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples:…

Given a point S (the light position) in P^3 and an algebraic surface Z (the mirror) of P^3, the caustic by reflection of Z from S is the Zariski closure of the envelope of the reflected lines got by reflection of the incident lines (Sm) on…

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten…

In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are…

For $p,q\ge2$ the $\{p,q\}$-tiling graph is the (finite or infinite) planar graph $T_{p,q}$ where all faces are cycles of length $p$ and all vertices have degree $q$. We give algorithms for the problem of recognizing (induced) subgraphs of…

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the…

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $\mathcal{P}=\{P_{u,v}|u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices…

We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to…

For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…

We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon) bi-Lipschitz homeomorphism between S'_1 and…

We establish a uniformization result for metric surfaces - metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be…

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of $\Sf^n\times \R$, extending the classification of umbilical surfaces in $\Sf^2\times \R$ by Rabah-Souam and Toubiana as well as the local…

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of…