arXiv research

Lie minimal surfaces are characterized by differential equations of principal curvatures.

In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

The article constructs helicoidal and catenoidal minimal surfaces in a Lie group.

New minimal surfaces found in a specific type of 3D space.

In this paper we will show the existence and uniqueness of the solution of the Björling problem for minimal surfaces in a 3-dimensional Lorentzian Lie group.

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.

We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.

Study on totally real flat minimal surfaces in quaternionic projective space.

The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

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 Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…

We prove existence and uniqueness of the solution of the Björling problem for minimal surfaces in a three-dimensional Lie group.

We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfac…

This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is a…

We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces …

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for s…

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 construct two one-parameter families of minimal properly embedded surfaces in the Lie group Sol3 using a Weierstrass-type representation. These surfaces are not invariant by a one-parameter group of ambient isometries. The first one can be viewed as a family of helicoids, and the second one is a family of minimal an…

Paper bounds the sum of index and nullity of minimal surfaces in certain 3-manifolds.

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for $k$-symmetric spa…

Unified rigidity theorem for cyclic and alternating surfaces.

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the cla…

In this paper we study some geometric properties of surfaces in the Heisenberg group, $\mathcal{H}_{3}.$ We obtain, using the Gauss map for Lie groups, a partial classification of minimal graphs in $\mathcal{H}_{3}.$ We also proof the non existence of umbilical surfaces in $\mathcal{H}_{3}.$

This article proves that if M is a smooth manifold of dimension at least four, then for generic choice of metric on M, all prime parametrized minimal surfaces in M are free of branch points and lie on nondegenerate critical submanifolds for the two-variable energy function which have the same dimension as the group of …

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral repre…

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation $H+σ=0$, where $H$ is analogous to the…

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

New minimal surface theory disproves a conjecture in symmetric spaces.

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depend…

New minimal surfaces in spheres with complex topologies from capillarity.

The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.

Constructs minimal surfaces near the boundary of a ball.

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…

Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as th…

Paper studies complex Lagrangian surfaces and their relation to $\mathrm{SL}(3,\mathbb{C})$-representations.

Study on subgroups' evolution in 3D Lie groups using mean curvature flow.

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…

We classify non-minimal biconservative surfaces with parallel mean curvature vector field in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$. When these surfaces do not lie in $\mathbb{S}^n$ or $\mathbb{H}^n$ and they are not vertical cylinders, we find their explicit (local) equation. We also prove a…

This note provides some new perspectives and calculations regarding an interesting known family of minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$. The surfaces in this family are the catenoids, parabolic catenoids and tall rectangles. Each is foliated by either circles, horocycles or circular arcs in horizontal c…

Curved flats linked to pairs of Lie applicable surfaces.

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

Criterion for Lie algebroid connections on compact Riemann surfaces.

We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over $\R$ and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.

The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.