Related papers: Uniqueness of the stereographic embedding

We prove the existence and uniqueness of geometric models of local isometry classes of locally homogeneous spaces with sectional curvature $|\operatorname{sec}|\leq 1$. Moreover, we show that the set of geometric models is compact in the…

Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a…

We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they…

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated…

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be…

In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber \cite{Hu57} for complete open surfaces to general dimensions based on the n-Laplace equations in conformal geometry. We are…

Suppose $M$ is a compact Riemannian manifold and $p\in M$ an arbitrary point. We employ estimates on the volume growth around $p$ to prove that the only conformal compactification of $M\setminus\{p\}$ is $M$ itself.

It has been shown that spaces of geodesic triangulations of surfaces with negative curvature are contractible. Here we propose an approach aiming to prove that the spaces of geodesic triangulations of a surface with negative curvature are…

We study the reflectional symmetry of a surface in the Euclidean 3-dimensional space with a cross-cap singularity with respect to planes. This symmetry is picked up by the singularities of folding maps on the cross-cap. We give a list of…

This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a ``weighted'' higher order mean curvature. Precisely, we show that a compact hypersurface $\Sigma^{n-1}$ embedded in $\H^n$ with $VH_k$ being…

We discuss a notion of discrete conformal equivalence for decorated piecewise euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle…

We present a constructive approach for approximating the conformal map (uniformization) of a polyhedral surface to a canonical domain in the plane. The main tool is a characterization of convex spaces of quasiconformal simplicial maps and…

We discuss a 4d Lagrangian descriptions, across dimensions IR dual, of compactifications of the 6d $(\text{D},\text{D})$ minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a…

Euclidean conformal integrals for an arbitrary number of points in any dimension are evaluated. Conformal transformations in the Euclidean space can be formulated as the Moebius group in terms of Clifford algebras. This is used to interpret…

For $d\in\mathbb{N}$, a compact sphere packing of Euclidean space $\mathbb{R}^{d}$ is a set of spheres in $\mathbb{R}^{d}$ with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous…

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic…

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral…

Let $M=G/H$ be a compact, simply connected, Riemannian homogeneous space, where $G$ is (almost) effective and $H$ is a simple Lie group. In this paper, we first classify all $G$-naturally reductive metrics on $M$, and then all $G$-geodesic…

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by…