A note on the Gauss map of complete nonorientable minimal surfaces

We construct complete nonorientable minimal surfaces whose Gauss map omits two points of the projective plane. This result proves that Fujimoto's theorem is sharp in nonorientable case.Comment: 8 pages, to appear in Pacific J. Mat

Compact spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime with a non-degenerate lightlike normal direction

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $\eta$. Several sufficient assumptions on such a surface with non-degenerate $\eta$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With this aim, a new formula which relates the Gauss curvatures of the induced metric and of the $\eta$-second fundamental form is developed. Then, totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone such that its $\eta$-second fundamental form is non-degenerate and has constant Gauss curvature two. Another characterizations of totally umbilical round spheres in terms of the Gauss-Kronecker curvature of $\eta$ and the area of the $\eta$-second fundamental form are also given

A certifying and dynamic algorithm for the recognition of proper circular-arc graphs

We present a dynamic algorithm for the recognition of proper circular-arc (PCA) graphs, that supports the insertion and removal of vertices (together with its incident edges). The main feature of the algorithm is that it outputs a minimally non-PCA induced subgraph when the insertion of a vertex fails. Each operation cost $O(\log n + d)$ time, where $n$ is the number vertices and $d$ is the degree of the modified vertex. When removals are disallowed, each insertion is processed in $O(d)$ time. The algorithm also provides two constant-time operations to query if the dynamic graph is proper Helly (PHCA) or proper interval (PIG). When the dynamic graph is not PHCA (resp. PIG), a minimally non-PHCA (resp. non-PIG) induced subgraph is obtained.Comment: 44 pages, 8 figures, appendix with 11 pages and many figure

A Note On Polar Representations

We show how to determine a possibly reducible polar representation of a compact connected Lie group G from its history and dimension

Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.Comment: 16 pages, 3 figure

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs

We consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version, a proper circular-arc (PCA) model $\cal M$ is given and the goal is to obtain an equivalent UCA model $\cal U$. We show a linear time algorithm with negative certification that can also be implemented to run in logspace. In the bounded version, $\cal M$ is given together with some lower and upper bounds that the beginning points of $\cal U$ must satisfy. We develop a linear space $O(n^2)$ time algorithm for this problem. Finally, in the minimal version, the circumference of the circle and the length of the arcs in $\cal U$ must be simultaneously as minimum as possible. We prove that every UCA graph admits such a minimal model, and give a polynomial time algorithm to find it. We also consider the minimal representation problem for UIG graphs. As a bad result, we show that the previous linear time algorithm fails to provide a minimal model for some input graphs. We fix this algorithm but, unfortunately, it runs in linear space $O(n^2)$ time. Finally, we apply the minimal representation algorithms so as to find the minimum powers of paths and cycles that contain a given UIG and UCA models, respectively.Comment: 33 pages, 3 figure

Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in R3\mathbb{R}^3

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case.Comment: 24 pages, 3 figures, research article. This updated version introduces considerably simplifications of notations and arguments, and includes some improvements of the results. The paper will appear in the Transactions of the American Mathematical Societ

Low Dimensional Polar Actions

Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine which of these polar actions admit an invariant metric with non-negative curvature.Comment: Accepted for publication in Geometria Dedicata. Minor corrections from original version, 37 pages, 8 figure

New quantum (anti)de Sitter algebras and discrete symmetries

Two new quantum anti-de Sitter so(4,2) and de Sitter so(5,1) algebras are presented. These deformations are called either time-type or space-type according to the dimensional properties of the deformation parameter. Their Hopf structure, universal R matrix and differential-difference realization are obtained in a unified setting by considering a contraction parameter related to the speed of light, which ensures a well defined non-relativistic limit. Such quantum algebras are shown to be symmetry algebras of either time or space discretizations of wave/Laplace equations on uniform lattices. These results lead to a proposal fortime and space discrete Maxwell equations with quantum algebra symmetry.Comment: 10 pages, LaTe

Total 2-domination of proper interval graphs

A set of vertices $W$ of a graph $G$ is a total $k$-dominating set when every vertex of $G$ has at least $k$ neighbors in $W$. In a recent article, Chiarelli et al.\ (Improved Algorithms for $k$-Domination and Total $k$-Domination in Proper Interval Graphs, Lecture Notes in Comput.\ Sci.\ 10856, 290--302, 2018) prove that a total $k$-dominating set can be computed in $O(n^{3k})$ time when $G$ is a proper interval graph with $n$ vertices and $m$ edges. In this note we reduce the time complexity to $O(m)$ for $k=2$.Comment: 8 pages, 4 figure