Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.
problem Exploring conditions for maximality of Hilbert square of real surfaces.
method Analyzing Hilbert square of maximal real surfaces and examining specific examples.
result Hilbert square can be maximal even for surfaces with disconnected real locus.
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.
Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parame…
We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show that they satisfy an Osserman-type inequality.
Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to re…
Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.
problem Characterizing maximal surfaces in Lorentz-Minkowski space.
method Three proofs showing correspondence to minimal surfaces in Euclidean space.
result Conjugate surface of a maximal graph over a convex domain is also a graph.
As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment L, the surface is invariant under the 180∘-rotation with respect to L. However, such a reflection property…
The paper classifies maximal translation surfaces in Lorentz-Minkowski space.
problem Classifying maximal translation surfaces in Lorentz-Minkowski space.
method Analyzing surfaces defined as the sum of two spatial curves, proving properties of generating curves, and classifying surfaces based on curve types.
result A full description of maximal translation surfaces, including new examples not found in Euclidean space.
The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.
problem Decomposing the height function of different types of surfaces into simpler components.
method Using Euler-Ramanujan identities and Weierstrass-Enneper representation to decompose height functions of minimal, maximal, timelike minimal, and Born-Infeld surfaces.
result The height function of various surfaces can be expressed as a finite sum of scaled and translated versions of itself.
In the present paper we study two-dimensional maximal surfaces with harmonic level-sets. As a corollary we obtain a new class of one-periodic maximal surfaces.
Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, w…
Discrete maximal surfaces identified from s-embeddings.
problem Understanding the conformal invariance of the Ising model.
method Introduced a special class of isothermic s-embeddings that correspond to discrete S-maximal surfaces.
result Each S-maximal surface comes with a 1-parameter family of associated surfaces that are isometric.
Maximal dilatation found on nonorientable surfaces.
problem Finding maximal dilatation on nonorientable surfaces.
method Proving irreducibility of a polynomial to show maximal dilatation.
result Maximal dilatation is achieved by the Liechti-Strenner polynomial.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
The paper studies a new class of affine maximal surfaces with singularities.
problem Understanding the properties of affine maximal surfaces with singularities.
method Defining a new subclass of affine maximal surfaces and applying Euclidean minimal surface theory.
result Affine maxfaces satisfy an Osserman-type inequality and do not contain non-trivial improper affine fronts.
Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
problem Understanding polygonal surfaces in pseudo-hyperbolic spaces.
method Characterizes polygonal surfaces by total curvature finiteness and asymptotic flatness, using comparison of ideal boundaries.
result Polygonal surfaces have parabolic type and polynomial quartic differential.
The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence, some existence and uniqueness results for maximal Moebius strips and maximal Klei…
Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.
problem Finding maximal surfaces with given boundary curves in pseudo-hyperbolic spaces.
method Defined and proved the existence of unique solutions using asymptotic Plateau problem and analysis of pseudo-holomorphic curves.
result Existence and uniqueness of maximal surfaces with specified boundary conditions.
New findings on hypersurfaces in Euclidean space that are both maximal and minimal.
problem Characterizing hypersurfaces in Euclidean space that are both maximal and minimal.
method Analyzing the level curves of the hypersurfaces and showing they are minimal hypersurfaces in the lower-dimensional Euclidean space.
result The level curves of these hypersurfaces are minimal hypersurfaces in the lower-dimensional Euclidean space.
A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold N is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in ł3. In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …
A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter space-time are obtained in order to illustrate this correspondence.
Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.
problem Characterizing and understanding group actions on surfaces, especially maximal handlebody and Hurwitz groups.
method Analyzing various group actions, comparing Hurwitz and handlebody groups, and examining bounding actions.
result Relationship between Hurwitz groups and maximal handlebody groups, and insights into geometric bounding actions.
Study on loops on non-orientable surfaces, determining cardinality and order.
problem Determining the cardinality and order of maximal complete 1-systems of loops on non-orientable surfaces.
method Proved the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is 2∣χ∣(∣χ∣+1), and used this to determine the cardinality of maximal complete 1-systems of loops. result Exact cardinality of maximal complete 1-systems of loops on punctured projective planes is determined.
We study area-stationary, or maximal, surfaces in the space L(H3) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L(H3) is a maximal surface. We then classify Lagrangian maximal surfa…
We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space L3 that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in L3. There exists a linear transformation between such a maximal surface and …
The study connects minimal and maximal surfaces in 3D and 3-L space.
problem Describing correspondences between minimal and maximal surfaces in different spaces.
method Weierstrass representation and asymptotic analysis.
result Established criteria for singularity types and moduli spaces.
Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces.
problem Proving uniqueness of measure of maximal entropy for geodesic flows on surfaces.
method Analyzes geodesic flows on closed orientable C^∞ surfaces, proving uniqueness of measure of maximal entropy and at most one SRB measure.
result Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces, covering previous results and new examples.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. Maximal representations into SO0(2,3) have bounded volume.
problem Bounding the volume of maximal representations into SO0(2,3). method Uniform upper and lower bounds on the volume for different surface groups.
result Volume is bounded from above and below for maximal representations into SO0(2,3). We use the Björling problem in Lorentz-Minkowski space to obtain explicit parametrizations of maximal surfaces containing a circle and a helix. We investigate the Weierstrass representation of these surfaces.
We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further…
Constructs graphs with singularities in a special space.
problem Creating graphs with specific singularities in a unique space.
method Using Weierstrass representation for minimal surfaces.
result Constructs entire singly periodic graphs with isolated cone-like singularities.
Interpolates curves using maximal and minimal surfaces in different spaces.
problem Interpolating curves in spacelike and Euclidean spaces.
method Using maximal and minimal surfaces, interpolates curves based on the Björling problem.
result Constructs surfaces to interpolate curves in a specified manner.
Bour's minimal surface has remarkable properties in three dimensional Minkowski space. We reveal the definite and indefinite cases of the Bour's surface using Weierstrass representations, and give some differential geometric properties of the astonishing maximal and minimal surfaces.
In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form M2×R1, where M2 is a connected Riemannian surface and M2×R1 is endowed with the Lorentzian metric <,>=<,>M−dt2. In particular, when M is a Riem…
This paper answers a question about discrete embeddings to maximal surfaces.
problem The question of whether discrete embeddings lift to maximal surfaces.
method Introduced a correspondence between s-embeddings and congruences of touching Lorentz spheres, identified isothermic s-embeddings that lift to S-isothermic surfaces.
result Isothermic s-embeddings lift to S-isothermic surfaces, which are key for obtaining discrete maximal surfaces.
The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.
We prove that a maximal surface in Lorentz-Minkowski space L3 can be extended analytically along its boundary if the boundary lies in a plane meeting the surface at a constant angle.
Researchers prove existence of metrics maximizing Laplace eigenvalue on all closed surfaces.
problem Proving the existence of metrics maximizing the first Laplace eigenvalue on closed surfaces.
method By contradiction and refinement of techniques, proving strict monotonicity under surface modifications.
result Existence of metrics maximizing the area-normalized first eigenvalue on all closed surfaces.
It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to such parameters the surface is determined uniquely up to a motion by the Gauss …
The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maxima…
Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.
problem Maximally symmetric (2,3,5)-distribution of An-Nurowski surface rolling without slipping or twisting. method Calculated vector fields defining a split g2 Lie algebra and projected to an action of SL(3,R). result Obtained an action of SL(3,R) on the configuration space without a surface. New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than π. We interpret this result in terms of Teichmüller theory, and prove the existence of …
Study maximal representations of surface groups via pleated surfaces in pseudo-Riemannian space.
problem Maximal representations of surface groups and their geometric properties.
method Introduction of ρ-invariant pleated surfaces and construction of shear cocycles. result Properties of ρ-invariant pleated surfaces, including embeddedness, acausality, and hyperbolic structure. Optimizes eigenvalues on surfaces with symmetries.
problem Maximizing Laplace and Steklov eigenvalues on Riemann surfaces with symmetries.
method Simplifies existing techniques for conformal class optimization.
result Proves existence and regularity of maximizers for Laplace and Steklov eigenvalues.
Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.
problem Characterizing dense points in the Bers boundary of Teichmüller space for infinite-type surfaces.
method Quasiconformal deformations and analytic conditions.
result Maximal cusps cannot approach certain points on the Bers boundary.