arXiv research

Virtual reality explores non-Euclidean Sol geometry.

The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.

The paper classifies hypersurfaces in a specific 4D geometry.

3-manifolds with specific homology are cobordant if homeomorphic.

Survey on four-dimensional Thurston geometries with Riemannian metrics.

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$. The maximal compact subgroup of $Isom(Sol_1^4)$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold with $Sol_1^4$-geometry whose holonomy is $D_4$. This implies that all …

Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.

Interpolates Sol geometry to Hyperbolic Space with a parameter.

Geodesics in Sol geometry described with invariant k and spiral properties.

In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Euclidean space into Sol …

In this paper we classify all surfaces in the 3-dimensional Lie group $Sol_3$ whose normals make constant angle with a left invariant vector field.

In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometr…

3-manifold groups have a property that allows them to act on quasi-trees.

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CM…

Let $G$ be a connected Lie group and $Γ\subset G$ a lattice. Connection curves of the homogeneous space $M=G/Γ$ are the orbits of one parameter subgroups of $G$. To $block$ a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connecting curve joining $m_1$ and $m…

We construct new explicit proper biharmonic functions on the $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.

In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We obtain a flux formula and we establish results that assert that, under some restr…

For any positive natural number $r\in\N^+$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the…

Let $Γ$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $Γ$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by enumerating $G$ by a regular language. However, in contrast to most earlier proofs of thi…

Three geometric analysis results on curve flows and Lie groups.

The paper classifies harmonic and biharmonic submersions from Sol space.

The Bonnet problem is solved for specific Thurston geometries.

A universal branched 3-manifold $W$ characterizes Sol 3-manifolds.

$Sol_3$ is shown not to be parabolic.

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds ap…

We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of th…

The paper studies geometric structures in Sol_3 with two connections.

Estimate sphere area in Sol group up to a factor of 10.

In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=α(s)\astβ(t)$, where $α$ and $β$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this paper we study translation surfaces in Sol$_3$ whose mean curvature vanishes.

Formula found for surfaces in Sol_3, leading to gap results.

Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.

Study on achiral Sol 3-manifolds with density results.

SOL is an open-source library for scalable online learning algorithms, and is particularly suitable for learning with high-dimensional data. The library provides a family of regular and sparse online learning algorithms for large-scale binary and multi-class classification tasks with high efficiency, scalability, porta…

Study on triharmonic curves in Sol space with constant curvature and torsion.

In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite …

In this paper, we give complete classifications of linear $\infty$-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all $\infty$-harmonic linear endomorphisms of Sol space and show that there is a subgroup of $\infty$-harmonic linear automorphisms in the group of linea…

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric…

The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

The paper describes distances on Sol-type groups using novel geometric techniques.

We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow.

In three dimensions, a `master theory' for all Thurston geometries requires imaginary flux. However, these geometries can be obtained from physical three-dimensional theories with various additional scalar fields, which can be interpreted as moduli in various compactifications of a higher-dimensional `master theory'. T…

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $φ:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on…

The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift …

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 some entire minimal graph G, then S is the image of G by a vertical translation…

The study determines $\mathbb{Z}_2$-Thurston norms in Sol manifolds and embeds non-orientable surfaces.