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.
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…
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.
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1…
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.
Study on totally real flat minimal surfaces in quaternionic projective space.
problem Characterizing the moduli space of totally real flat minimal immersions in HP^3.
method Analyzing the moduli space of linearly full totally real flat minimal immersions from C into HP^3.
result The moduli space has three components, each a 6-dimensional manifold.
The study constructs minimal submanifolds in complex and quaternionic projective spaces.
problem Finding minimal submanifolds in complex and quaternionic projective spaces.
method Using complex-valued harmonic morphisms.
result Complete minimal submanifolds of odd-dimensional complex projective spaces and their dual hyperbolic spaces are constructed.
Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.
problem Understanding the Morse index of minimal hypersurfaces in real projective spaces.
method Analyzing unstable one-sided and two-sided minimal hypersurfaces in real projective spaces.
result The Morse index of minimal hypersurfaces is at least n+2, with specific examples provided.
Paper proves unique energy-minimizing curves in constrained spaces.
problem Uniqueness of energy-minimizing curves in constrained spaces.
method Investigated energy-minimizing curves with fixed endpoints in a constrained space.
result Proved that the set of points for which the energy-minimizing curve is not unique has no interior points.
Constructs minimal submanifolds in symmetric spaces using eigenfunctions.
problem Finding minimal submanifolds in symmetric spaces.
method Employing recent results from S. Gudmundsson and T.J. Munn, constructing submanifolds using eigenfunctions.
result Constructs minimal submanifolds of classical compact Riemannian symmetric spaces.
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.
In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.
Study of minimal surfaces in 3D space with special connections.
problem Classification of minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
method Analysis of singular minimal translation surfaces in a 3D Euclidean space with a semi-symmetric connection.
result Classification of singular minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
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. Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Minimal surfaces found in 4D space.
problem Minimal surfaces in 4D space.
method Reduced biharmonic equation to ODEs, excluded non-minimal solutions.
result Biharmonic rotational surfaces in 4D are minimal.
The paper studies minimal submanifolds with specific curvature properties in Euclidean space.
problem Minimal submanifolds with (n−2)-umbilical properties in Euclidean space. method Established a correspondence and developed a Weierstrass type method for local parametrization.
result Minimal, generic, (n−2)-umbilic submanifolds are (n−2)-rotational and have a parametric description. Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
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.
A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquel…
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.
We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
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.
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k,2k−1) and the generalised quaternionic space S3/Q4k have complexity $k,…
Flat minimal hypersurfaces in 4D space are always flat.
problem Understanding stable minimal hypersurfaces in 4D space.
method Proving stability and completeness lead to flatness.
result Complete, stable minimal hypersurfaces in 4D are flat.
A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.
problem Extending the half-space theorem to higher dimensions in R4. method Analyzing the topological constraints on minimal hypersurfaces in R4. result A complete, properly embedded minimal hypersurface in a slab in R4 must be a hyperplane. Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
Totally geodesic submanifolds in hyperbolic space up to codimension two.
problem Characterizing minimal homogeneous submanifolds in hyperbolic spaces.
method Analyzing properties of minimal submanifolds in hyperbolic spaces up to codimension two.
result Minimal homogeneous submanifolds of hyperbolic space up to codimension two are totally geodesic.
New theorem finds new minimal hypersurfaces in hyperbolic space.
problem Finding new minimal hypersurfaces in hyperbolic space.
method Developed a min-max theory for complete minimal hypersurfaces.
result Shows existence of a new minimal hypersurface between two given ones.
A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…
The study proves that certain minimal hypersurfaces in 4D space must be planes.
problem The extension of the half-space theorem to higher dimensions is obstructed.
method Analyzes topological properties of minimal hypersurfaces in R4. result Complete, properly embedded minimal hypersurfaces in R4 with bounded curvature and diffeomorphic to R3 must be planes. Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
Minimal partitions with minimal perimeter found in metric spaces.
problem Finding minimal partitions with minimal perimeter in metric spaces.
method Existence proof and regularity analysis of minimal domains.
result Existence and regularity of minimal partitions in various metric spaces.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in S4 and H4. New parametrizations for minimal timelike surfaces discovered.
problem Finding parametrizations for minimal timelike surfaces in specific spaces.
method Derived representation formulas for null curves leading to parametrizations of minimal timelike surfaces.
result Examples of minimal timelike surfaces constructed.
The paper finds the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
problem Finding the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
method Using the Generalized Maximum Principle, the paper proves that a 3D complete minimal hypersurface with constant scalar curvature in H4(−1) satisfies S≤2921. result A 3D complete minimal hypersurface in H4(−1) with constant scalar curvature satisfies S≤2921. Sharp area estimates for minimal submanifolds in curved spaces.
problem Estimating the area of minimal submanifolds passing through a specific point.
method Proving sharp area estimates in hyperbolic and spherical spaces.
result Sharp area estimates analogous to Euclidean settings.
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …