Related papers: Analytic saddle spheres in $\mathbb{S}^3$ are equa…

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical…

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group over a regular semilocal ring is itself trivial. Extending the work of \v{C}esnavi\v{c}ius and Fedorov, we prove a non-noetherian…

For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is…

To, say, a proper algebraic or holomorphic space $X/S$, and a coherent sheaf ${\mathcal F}$ on $X$ we identify a functorial ideal, the fitted flatifier, blowing up sequentially in which leads to a flattening of the proper transform of…

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of…

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…

The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times…

We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the 2-sphere are analytic. This is a real analog for…

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space is a round sphere, provided its mean curvature and the norm of its position vector have an upper…

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the…

We produce examples of solutions to the non-abelian gravitating vortex equations, which are a dimensional reduction of the K\"aher-Yang-Mills- Higgs equations. These are equations for a K\"ahler metric and a metric on a vector bundle. We…

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to…

A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere.…

We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A…

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point…

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be…

Let $X$ be a germ of real analytic vector field at $({\mathbb R}^{2},0)$ with an algebracally isolated singularity. We say that $X$ is a topological generalized curve if there are no topological saddle-nodes in its reduction of…

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any…

We study the spectral metric aspects of the standard Podles sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dabrowski and Sitarz. The Dirac operator of…

Inspired by the equatorial concentration of measure phenomenon in the sphere, a result which is deduced from the general, (and intrinsic), concentration of measure in $\mathbb{S}^n(1)$, we describe in this paper an equatorial concentration…