Study on minimal surfaces in a 3D space with 2m-norm.
problem Characterizing minimal surfaces in a specific geometric space.
method Examining translation, homothetical, and separable minimal surfaces.
result New insights into minimal surfaces in a 3D space with 2m-norm.
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.
problem Constructing minimal surfaces in 3-sphere using reflections.
method Minimal n-gon solves free boundary problem; curvature lines combinatorics investigated. result New examples of minimal reflection surfaces based on pentagons.
Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.
problem Characterizing minimal surfaces in Kropina 3D space.
method Solving partial differential equations to characterize minimal surfaces.
result Only planes are minimal translation surfaces in Kropina 3D space.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
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.
New inequality helps map stability in minimal surfaces.
problem Stability of minimal surfaces in Rn. method Developing new inequalities and perspectives on minimal surfaces.
result Reproves instability of classical minimal surfaces like Enneper.
Minimal surfaces help compare geometric shapes.
problem Comparing different geometric shapes.
method Applications of minimal surfaces.
result New insights into geometric comparisons.
New minimal surfaces found using Toda lattice and integrable systems.
problem Constructing new minimal surfaces with specific genus.
method Using Toda lattice and integrable systems techniques.
result New singly periodic minimal surfaces with genus j(j+1)/2−1. This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Maximal surfaces in L3 correspond to timelike minimal surfaces.
problem Establishing a correspondence between maximal and timelike minimal surfaces in L3. method Linear transformation between maximal surfaces and timelike minimal surfaces, preserving singularities and Gauss map.
result One-one correspondence and preservation of properties between maximal and timelike minimal surfaces.
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
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…
Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.
problem Exploring singularities of timelike minimal surfaces in Minkowski 3-space.
method Existence and non-existence theorems, criteria for specific singularities.
result Various singularities unique to timelike minimal surfaces, including cuspidal butterfly and (2,5)-cuspidal edge. This paper solves minimal surface equations near Hardt-Simon foliations.
problem Minimal surfaces near Hardt-Simon foliations.
method Uses gluing methods to construct minimal surfaces.
result Constructs minimal surfaces over Hardt-Simon surfaces and near quadratic cones.
Explains minimal surfaces and their properties.
problem Understanding minimal surfaces and their characteristics.
method Analyzes various minimal surfaces and their properties.
result Discusses the properties and stability of minimal surfaces.
We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …
We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. …
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.
Solves minimal surface problem in H^2xR.
problem Minimal surfaces in H^2xR with specified asymptotic boundary.
method Identifies finite Jordan curves bounding minimal surfaces.
result Identifies specific Jordan curves for minimal surfaces.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. Study on minimal surfaces in Heisenberg group with duality formula.
problem Understanding minimal surfaces in Heisenberg group.
method Introducing transformation surfaces and using logarithmic derivative of moving frame.
result Derivation of Sym formula for dual minimal surface.
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.
Minimal surfaces are the only biharmonic in Sol3.
problem Characterizing biharmonic surfaces in Sol3.
method Found local equations for biconservative surfaces and showed all biharmonic surfaces are minimal.
result All biharmonic surfaces in Sol3 are minimal.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
We prove a general fusion theorem for complete orientable minimal surfaces in R3 with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal …
Minimal surfaces can be transformed into others with unchanged bending content.
problem Understanding the deformation properties of minimal surfaces.
method Refined polar decomposition theorem to identify bending-neutral deformations.
result Every minimal surface can be transformed into another by a bending-neutral deformation.
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.
The paper proves cylindrical nature of singular minimal ruled surfaces.
problem Understanding minimal potential energy surfaces under gravitational forces.
method Analyzing singular minimal ruled surfaces in Euclidean and Lorentz-Minkowski 3-spaces.
result Singular minimal ruled surfaces are cylindrical, including as α-catenary cylinders.
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.
Minimal surfaces in hyperbolic and anti-de Sitter spaces constructed using loop group methods.
problem Constructing minimal surfaces in hyperbolic and anti-de Sitter spaces.
method Loop group factorization methods.
result Minimal surfaces in hyperbolic and anti-de Sitter 3-space with n-punctured sphere topology. 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 grow area very quickly.
problem Understanding minimal surfaces with rapid area growth.
method Examples of minimal immersions in Euclidean space.
result Proper minimal surfaces with rapid area growth found.
We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…
Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.
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.
The study characterizes minimal surfaces in a specific three-dimensional metric space and finds that planes are the only minimal surfaces.
problem Characterizing minimal surfaces in a specific three-dimensional metric space.
method Obtained partial differential equations to characterize minimal surfaces and proved that planes are the only such surfaces.
result Planes are the only minimal surfaces in the Matsumoto space.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
We classify ruled minimal surfaces in R3 with density ez. It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in R3 with density ez. It is also proved that all translation minimal surfaces are ruled.
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.
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
Paper finds flat minimal surfaces in quaternionic projective spaces.
problem Characterizing flat minimal surfaces in quaternionic projective spaces.
method Analyzing totally real minimal surfaces of isotropy order n.
result Linearly full totally real flat minimal surfaces are two surfaces in C^n, one of which is the Clifford solution.