Describes rendering scenes in Nil geometry.
problem None explicitly stated in the abstract.
method Expository account of rendering real-time scenes in Nil geometry.
result Interesting geometric phenomena observed.
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48 results for “Nil_3”
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.
problem Classifying hypersurfaces in Nil^4.
method Using Lie group structure and Codazzi conditions.
result Characterization and classification of minimal hypersurfaces in Nil^4.
The Borsuk-Ulam theorem is applied to 3-manifolds with Nil geometry.
problem Determining involutions on 3-manifolds with Nil geometry.
method Applying the Borsuk-Ulam theorem to closed, connected 3-manifolds with Nil geometry.
result All free involutions on the 3-manifolds and their Borsuk-Ulam index are determined.
The paper studies deformations of Kundt metrics using nil-Killing vector fields.
problem Deformations of Kundt metrics in the direction of type III tensors.
method Characterizations within the Kundt class using nil-Killing vector fields.
result Theorem classifying algebraic stability of tensors and sufficient criteria for preserving spi's.
We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses contro…
The study examines closed manifolds with ray nil-affine structures and their completeness.
problem Characterizing closed manifolds with ray nil-affine structures.
method Analyzing properties of nilpotent spaces and flag manifolds, applying rank conditions and automorphism group actions.
result Closed manifolds are either complete or their developing map is a cover onto the complement of a nil-affine subspace.
We study Farrell Nil-groups associated to a finite order automorphism of a ring . We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite mu…
We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil.
Study of minimal surfaces with non-trivial topology in Heisenberg group using loop group method.
problem Constructing and classifying minimal surfaces with non-trivial topology in the Heisenberg group.
method Generalized Weierstrass type representation and loop group method.
result Classification of equivariant minimal surfaces given by one-parameter subgroups.
Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.
problem Characterize geodesics on a specific nil-manifold.
method Analyze the projection of sub-Riemannian structure, describe geodesic flow dynamics, and estimate sub-Riemannian geodesics.
result Sharp bounds and estimates for sub-Riemannian geodesics.
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 …
The paper estimates heights of constant mean curvature graphs in specific spaces.
problem Estimating heights of constant mean curvature graphs in and .
method Height estimates for compact, constant mean curvature graphs in and .
result Announced a structure-type result for proper graphs defined on relatively compact domains.
A translation surface in the Heisenberg group is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group .
The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
problem Verifying the Farrell-Jones Conjecture for groups acting on trees.
method Analyzing acylindrical actions on simplicial trees and using the Farrell-Jones Conjecture.
result The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
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…
We prove that there are exactly Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in , with as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly specific hyperbolic kno…
New biharmonic functions found on Thurston geometries.
problem Finding biharmonic functions on Thurston geometries.
method Constructing explicit proper biharmonic functions.
result Explicit biharmonic functions constructed on various Thurston geometries.
Study on -preserving diffeomorphisms in Lorentzian manifolds.
problem Existence and properties of -preserving diffeomorphisms in Lorentzian manifolds.
method Analysis of Lie derivatives and nil-Killing vector fields to determine -preserving diffeomorphisms.
result Finite transitive Lie algebra of nil-Killing vector fields exists for Kundt spacetimes with constant scalar polynomial curvature invariants.
Minimal surfaces in Heisenberg group solved via Dirichlet problems.
problem Existence of minimal surfaces in Heisenberg group with balanced metric.
method Solving Dirichlet problems for minimal surface equation.
result Existence of complete properly embedded minimal surfaces.
We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the si…
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
Constructs new explicit proper r-harmonic functions on Thurston geometries.
problem Developing proper r-harmonic functions on specific 3D geometries.
method Explicit construction of new functions for Thurston geometries.
result Explicit construction of new proper r-harmonic functions on various Thurston geometries.
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
problem Classifying homogeneous hypersurfaces in specific 4D geometries.
method Analyzing isometry groups and applying classification techniques.
result Homogeneous hypersurfaces identified in , and .
We study properly discontinuous and cocompact actions of a discrete subgroup of an algebraic group on a contractible algebraic manifold . We suppose that this action comes from an algebraic action of on such that a maximal reductive subgroup of fixes a point. When the real rank of any simple subg…
The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$,…
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute t…
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the defo…
Nil happens. What about Sol?math.DG
We construct complete, finite volume, 4-dimensional manifolds with sectional curvature with cusp cross sections compact solvmanifolds.
The paper classifies 3D contact partially hyperbolic diffeomorphisms.
problem Classifying contact partially hyperbolic diffeomorphisms in 3D.
method Smooth classification, conjugation to known flows or automorphisms, use of invariant distributions.
result Classification up to finite quotient or power, conjugation to known structures.
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…
The paper classifies solitons in the Heisenberg space.
problem Classifying solitons in the Heisenberg space.
method Analysis of four canonical types of Killing vector fields and three families of invariant surfaces.
result Explicit parametrizations and geometric properties of solitons found.
Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.
problem Algebraic K-theory of 3-manifold groups.
method Farrell-Jones isomorphism conjecture, models for virtually cyclic subgroups, geometrization theorem.
result Descriptions of Whitehead groups and algebraic K-theory groups in terms of finite subgroups and Nil-groups.
Survey on four-dimensional Thurston geometries with Riemannian metrics.
problem Understanding homogeneous manifolds and their Riemannian metrics.
method Description and analysis of specific Thurston geometries in four dimensions.
result Exhibition of all Riemannian metrics invariant under group actions.
The paper classifies surfaces in the Heisenberg space invariant under specific isometries.
problem Classifying surfaces in the Heisenberg space with specific geometric properties.
method Analyzing surfaces with mean curvature under left-translations, rotations, and helicoidal motions.
result Classification of -translators invariant under specific isometries.
Study Lie algebras of indefinite spaces with finite volume.
problem Characterize isometry Lie algebras of indefinite homogeneous spaces.
method Analyze nil-invariant symmetric bilinear forms and their adjoint representations.
result Derive structure theorem and classify isometry algebras for finite volume.
A solution to the Bernstein problem in the three-dimensional Heisenberg group via loop groupsmath.DG
In this note we present a short alternative proof for the Bernstein problem in the three-dimensional Heisenberg group by using the loop group technique.
We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases and , the flow is qualitatively the same as the Ricci flow. In the cases and , if the curvature is small, t…
Constructs Lorentzian harmonic maps and associated timelike surfaces.
problem Lorentzian harmonic maps and timelike surfaces properties.
method Constructs framed null curves and solves eigenvalue equation.
result Characterizes singularities on timelike minimal surfaces.
We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbound…
3-manifold groups have a property that allows them to act on quasi-trees.
problem Proving property (QT) for 3-manifold groups.
method Analyzing the geometry of 3-manifolds and using quasi-trees.
result Compact, orientable 3-manifold groups have property (QT) if they don't support Sol or Nil geometry.
In this paper we find functions over bounded domains in the 2-dimensional Euclidean space, whose graphs (in the Heisenberg space) has constant mean curvature different from zero and taking on (possibly) infinite boundary values over the boundary of the domain.
For graphs with 13 or more vertices, either the graph or its complement is intrinsically linked.
problem Identifying graphs with 13 or more vertices that are intrinsically linked or their complements.
method Analyzing properties of graphs and their complements to determine if either is intrinsically linked.
result For graphs with 13 or more vertices, either the graph or its complement is intrinsically linked.
In this paper, we give complete classifications of linear -harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all -harmonic linear endomorphisms of Sol space and show that there is a subgroup of -harmonic linear automorphisms in the group of linea…
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…
We study -dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the -di…
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions…