The paper characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
problem Characterizing minimal surfaces and Lagrangian surfaces in complex projective space.
method Using Ruh-Vilms type theorems.
result Characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
This paper classifies minimal complex surfaces with Levi-Civita Ricci-flat metrics.
problem Classifying minimal complex surfaces with specific geometric properties.
method Study of compact complex manifolds with Levi-Civita Ricci-flat metrics.
result Minimal complex surfaces with Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces and Hopf surfaces.
Complex analysis aids in studying minimal surfaces.
problem Understanding minimal surfaces in Euclidean spaces.
method Complex-analytic techniques applied to conformal minimal surfaces.
result New results on approximation, interpolation, and general position properties.
Study minimal Lagrangian surfaces in complex projective plane, focusing on contractible cases.
problem Construct minimal Lagrangian surfaces in complex projective plane.
method Loop group method
result Presented new classes of minimal Lagrangian surfaces.
New minimal surfaces found with Cantor ends in convex domains.
problem Finding complex structures for minimal surfaces with Cantor ends.
method Proving existence of complete minimal surfaces with Cantor ends in minimally convex domains.
result Existence of a Cantor set whose complement forms a complete minimal surface.
Disk complexes show 3-sphere surfaces are topologically minimal.
problem Understanding minimal surfaces in 3-sphere topology.
method Analyzing disk complexes of genus >1 Heegaard surfaces.
result Genus >1 Heegaard surfaces have minimal index 2g-1.
Survey of complex analytic methods in minimal surface theory.
problem Global theory of minimal surfaces in Euclidean spaces.
method Complex analytic methods including Oka theory, holomorphic sprays, and Riemann-Hilbert boundary value problem.
result New constructions and results on minimal surfaces in various contexts.
Study compares different complexity criteria for free boundary minimal surfaces.
problem Comparing different complexity criteria for free boundary minimal surfaces.
method Global theory of free boundary minimal surfaces.
result Provides a complete picture of how area, topology, and Morse index compare.
Minimal genus surfaces solve homology problems in finite complexes.
problem Finding surface representatives of homology classes with minimal genus.
method Minimizing genus and Euler characteristic, analyzing surgeries and homotopy.
result Minimizers are homotopic to cellwise coverings, but problem is undecidable in general.
The paper studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
problem Characterizing timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
method Defining natural Gauss maps and proving a Ruh-Vilms type theorem.
result Timelike minimal Lagrangian surfaces correspond to the fifth real form of the complex affine Kac-Moody algebra.
Study minimal surfaces in complex hyperbolic space, linking entropy and volume.
problem Characterize minimal submanifolds in complex hyperbolic space.
method Analyze asymptotic regularity and introduce Colding-Minicozzi entropy and CR-volume.
result Establish a connection between Colding-Minicozzi entropy and CR-volume.
Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
Study models Ricci flow on complex surfaces, showing mixed behavior.
problem Understanding long-time behavior of Ricci flow on complex surfaces.
method BiLipschitz models for 4-manifolds (minimal surfaces of general type).
result Exhibits a combination of expanding and static behavior.
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
Develops a new method for minimal Lagrangian surfaces in complex quadrics.
problem Finding minimal Lagrangian surfaces in complex quadrics.
method Loop group representation and flat connections.
result Equivalence of minimality and flatness of connections, explicit families of examples.
Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on …
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
The study finds minimal surfaces in complex space forms are often totally geodesic.
problem Characterizing minimal surfaces with specific geometric properties in complex space forms.
method Analyzing free-boundary minimal surfaces in geodesic balls of complex space forms.
result Minimal surfaces in certain complex space forms are either totally geodesic or superminimal.
The paper provides Enneper representations for minimal surfaces in Lorentz-Minkowski space.
problem Finding representations for minimal surfaces in Lorentz-Minkowski space.
method Using complex and paracomplex analysis, the paper constructs Enneper representations for both spacelike and timelike minimal surfaces.
result Various examples of minimal surfaces in L3 are constructed using the Enneper representation formula. Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.
problem Characterize minimal timelike surfaces in R13. method Use a Weierstrass-type formula with holomorphic functions in split-complex numbers to find canonical parameters and corresponding holomorphic functions.
result Enneper surfaces are the only minimal timelike surfaces with polynomial parametrization of degree 3 in isothermal parameters.
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hami…
A general study of minimal surfaces of the Riemannian product of two spheres S^2xS^2 is tackled. We stablish a local correspondence between (non-complex) minimal surfaces of S^2xS^2 and certain pair of minimal surfaces of the sphere S^3. This correspondence also allows us to link minimal surfaces in S^3 and in the Riem…
The study finds that most minimal surfaces in generic 4D manifolds intersect in complex ways.
problem Understanding self-intersections of minimal surfaces in generic Riemannian manifolds.
method Analyzing the properties of minimal surfaces in a generic Riemannian manifold of dimension four.
result Most minimal surfaces in generic 4D manifolds intersect in complex ways, with tangent planes failing to be complex with respect to any orthogonal complex structure.
New minimal surfaces in 4D space discovered using complex rotations.
problem Discovering new minimal surfaces in 4D space.
method Complex parabolic rotations of holomorphic null curves in 4C space.
result Existence of minimal surfaces foliated by conic sections in 4D space.
Paper introduces Chern minimal surfaces in Hermitian surfaces and establishes identities related to their points and bundles.
problem Understanding the properties of Chern minimal surfaces in Hermitian surfaces.
method Using the Chern connection, the paper introduces Chern minimal surfaces and establishes identities related to their points and bundles.
result Established two identities relating the orders of complex and anticomplex points, the cap product of pull-back of first Chern class, and the Euler characteristics of tangent and normal bundles.
New stable minimal surfaces generalize classical Henneberg surface.
problem Finding new stable minimal surfaces in 3D.
method Generalized Henneberg surface with infinite families of complete, non-orientable surfaces.
result Infinite families of complete, finitely branched, non-orientable, stable minimal surfaces.
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…
Discretizes projective minimal surfaces using geometric characterizations.
problem Classifying discrete projective minimal surfaces.
method Introduced canonical discrete models and line congruences.
result Discrete analogues of classical Lie quadrics and surfaces.
The paper studies timelike minimal surfaces in De Sitter space using complex analysis.
problem Analyzing timelike minimal surfaces in De Sitter space.
method Complex variable analysis and stereographic projection.
result Explicit construction of many families of minimal timelike surfaces.
Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.
problem Constructing maps between Lagrangian submanifolds and quaternionic projective spaces.
method Explicit construction of maps from minimal δ(2)-ideal Lagrangian submanifolds of Cn to HPn−1. result One-to-one correspondences between minimal Lagrangian surfaces in CP2 and minimal totally complex surfaces in HP2. Researchers compute covering type of all closed surfaces.
problem Measuring the complexity of closed surfaces using covering type.
method Using the concept of covering type introduced by Karoubi and Weibel, the researchers computed the minimum number of vertices in simplicial complexes homotopy equivalent to closed surfaces.
result Results completely settle a problem posed by Karoubi and Weibel, and provide insights into the relationship between surface topology and minimal triangulations.
A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation between area minimizers and complex curves. When the Kaehler surface is a K3 surfa…
We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanif…
We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal t…
We study the problem of existence of F-structures on compact complex surfaces, giving a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces. For instance we prove that compact Kahler surfaces o…
The minimal surface equation Q in the second order contact bundle of R3, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form Omega on $Q\0$. The minimal surfaces M in R3 correspond to the complex analytic curves C in Q, where the derivati…
Study totally real flat minimal surfaces in hyperquadric.
problem Characterize totally real minimal surfaces in complex hyperquadric.
method Analyze geometric properties and use harmonic sequences.
result Classify totally real flat minimal surfaces for N=4, 5, 6.
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in Rn for any n≥3. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…
Canonical coordinates defined for minimal time-like surfaces in n-dimensional Minkowski space.
problem Characterizing canonical coordinates on minimal time-like surfaces.
method Introducing canonical coordinates and proving their existence and uniqueness; using analysis over the algebra of double numbers.
result Canonical coordinates on minimal time-like surfaces are characterized by a natural condition for a complex function over the algebra of double numbers.
We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which ar…
We construct harmonic diffeomorphisms from the complex plane C onto any Hadamard surface M whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in M×R over domains of M bounded by ideal geodesic polygons and show the existence of a se…
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.
Symplectic fillings of surface singularities linked to minimal model program.
problem Symplectic fillings of quotient surface singularities.
method Sequence of rational blow-downs and symplectic antiflips.
result Every minimal symplectic filling can be obtained from minimal resolution via rational blow-downs and antiflips.
The paper proves dense minimal surfaces in arbitrary domains of R^n.
problem Proving complete minimal surfaces in arbitrary domains of R^n.
method Proof of dense minimal surfaces using compact-open topology and adapted methods for non-orientable surfaces.
result Every domain in R^n contains complete minimal surfaces that are dense and have arbitrary orientable topology.
Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.
problem Finding minimal entropy diffeomorphisms on K3 surfaces.
method Constructs pseudo-Anosov diffeomorphisms minimizing entropy.
result Obtains infinitely many entropy-minimizing diffeomorphisms.