Maximal Surfaces Research Articles

Geometry of surfaces with singular points—applications to maximal surfaces in Lorentz-Minkowski space

Geometry of surfaces with singular points—applications to maximal surfaces in Lorentz-Minkowski space

Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space

abstract: We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\Sp(4,\R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $\Sp(4,\R)$-Higgs bundles over $\CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\h^{2,2}$ associated to $\Sp(4,\R)$-Hitchin representations along rays of holomorphic quartic differentials.

Bounded differentials on the unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r r -differentials. We study the relationship between bounded holomorphic r r -differentials/ ( r − 1 ) (r-1) -differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space S L ( r , R ) / S O ( r ) SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in R 3 \mathbb {R}^3 , maximal surfaces in H 2 , n \mathbb {H}^{2,n} and J J -holomorphic curves in H 4 , 2 \mathbb {H}^{4,2} . Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.

Calabi-Bernstein type results for critical points of a weighted area functional in [formula omitted] and [formula omitted]

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete [φ,e→3]-minimal surfaces in R3 whose Gauss maps lie on compacts subsets of open hemispheres of S2. We also give a general non-existence result for complete spacelike [φ,e→3]-maximal surfaces in L3 and, in particular, we obtain a Calabi-Bernstein type result when φ˙ is bounded.

Plateau problems for maximal surfaces in pseudo-hyperbolic space

Plateau problems for maximal surfaces in pseudo-hyperbolic space

Duality of boundary value problems for minimal and maximal surfaces

Duality of boundary value problems for minimal and maximal surfaces

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

We study in this paper quasiperiodic maximal surfaces in pseudo-hyperbolic spaces and show that they are characterised by a curvature condition, Gromov hyperbolicity or conformal hyperbolicity. We show that the limit curves of these surfaces in the Einstein Universe admits a canonical quasisymmetric parametrisation, while conversely every quasisymmetric curve in the Einstein Universe bounds a quasiperiodic surface in such a way that the quasisymmetric parametrisation is a continuous extension of the uniformisation; we give applications of these results to asymptotically hyperbolic surfaces, rigidity of Anosov representations and a version of the universal Teichmüller space.

Higher genus nonorientable maximal surfaces in the Lorentz-Minkowski 3-space

We study nonorientable maximal surfaces in Lorentz-Minkowski 3-space. We prove some existence results for surfaces of this kind with high genus and one end.

The algebraic surfaces of the minimal-maximal surfaces

Considering soft computing, the Weierstrass data (??1/2, ?1/2) gives two different minimal surface equations and figures. By using hard computing, we give the family of minimal and spacelike maximal surfaces S(m,n) for natural numbers m and n in Euclidean and Minkowski 3-spaces E3, E2,1, respectively. We obtain the classes and degrees of surfaces S(m,n). Considering the integral free form of Weierstrass, we define some algebraic functions for S(m,n). Indicating several maximal surfaces of value (m, n) are algebraic, we recall Weierstrass-type representations for maximal surfaces in E2,1, and give explicit parametrizations for spacelike maximal surfaces of value (m, n). Finally, we compute the implicit equations, degree, and class of the spacelike maximal surfaces S(0,1), S(1,1) and S(2,1) in terms of their cartesian or inhomogeneous tangential coordinates in E2,1.

Interior estimates for the Monge–Ampère type fourth order equations

In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Guti\'errez's estimate, including the Chern conjecture on affine maximal surfaces. For the inhomogeneous equation, we also obtain a new proof in dimension two by an integral method relying on the Monge-Amp\`ere Sobolev inequality. This proof works even when the right hand side is singular. In higher dimensions, we obtain the interior regularity in terms of integral bounds on the second derivatives and the inverse of the determinant.

Reflection Principles for Zero Mean Curvature Surfaces in the Simply Isotropic 3-space

Zero mean curvature surfaces in the simply isotropic 3-space $${\mathbb {I}}^3$$ naturally appear as intermediate geometry between geometry of minimal surfaces in $${\mathbb {E}}^3$$ and that of maximal surfaces in $${\mathbb {L}}^3$$ . In this paper, we investigate reflection principles for zero mean curvature surfaces in $${\mathbb {I}}^3$$ as with the above surfaces in $${\mathbb {E}}^3$$ and $${\mathbb {L}}^3$$ . In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $${\mathbb {I}}^3$$ , along which the induced metrics become singular.

Chern-Ricci harmonic functions on zero mean curvature surfaces in the three-dimensional Lorentz-Minkowski space and the rigidity of Enneper's surface

In this paper, we introduce Chern-Ricci functions on spacelike maximal surfaces and timelike minimal surfaces in L3, and we demonstrate that they are harmonic. Thereafter, we obtain a rigidity (uniqueness) result for classical examples of surfaces with zero mean curvature in L3 using the constancy of the Chern-Ricci functions. Particularly, we prove that Enneper's surface is the only surface that possesses zero mean curvature with a constant first Chern-Ricci function. We also prove that surfaces having zero mean curvature with a constant second Chern-Ricci function are contained in the moduli space of catenoids, helicoids, and their associated families up to homotheties.

Extension of Krust theorem and deformations of minimal surfaces

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space \(\mathbb {E}^3\) is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space \(\mathbb {L}^3\). In this article, we introduce a new deformation family that continuously connects minimal surfaces in \(\mathbb {E}^3\) and maximal surfaces in \(\mathbb {L}^3\), and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space \(\mathbb {I}^3\) appear in the middle of the above deformation family. We also prove another type of Krust’s theorem for this family, which implies that the graphness of such minimal surfaces in \(\mathbb {I}^3\) strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

Spinorial Representation of Surfaces in Lorentzian Homogeneous Spaces of Dimension $$\varvec{3}$$

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in \(\mathbb {L}(\kappa ,\tau )\) spaces. We then recover the Calabi correspondence between minimal surfaces in \(\mathbb {R}^3\) and maximal surfaces in \(\mathbb {R}_1^3\), and obtain a new Lawson type correspondence between CMC surfaces in \(\mathbb {R}_1^3\) and in the 3-dimensional pseudo-hyperbolic space \(\mathbb {H}_1^{3}.\)

Maximal simply connected Lorentzian surfaces with a Killing field and their completeness

In the first part of this paper, we give a global description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff manifold (the space of Killing orbits) and a smooth function defined there. In the second part, we study the completeness of such surfaces and prove in particular that under the hypothesis of bounded curvature, completeness is equivalent to null completeness. We also give completeness criterions involving the topological structure of the space of Killing orbits, or both the topology of this space and the geometry of the surface.

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζ∈C, m∈Z≥1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Maximal spacelike surfaces in a certain homogeneous Lorentzian 3-manifold

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds, which includes Minkowski 3-space and anti-de Sitter 3-space, 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 representation formula for maximal spacelike surfaces in the homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of maximal spacelike surfaces and its harmonicity are discussed.

Quantum phase diagrams of matter-field Hamiltonians I: Fidelity, Bures distance, and entanglement

A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.

Representation of Maximal Surfaces in a 3-Dimensional Lightlike Cone

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone Q3 is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via the representation formula.