Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
Computes minimal polynomials for generalized Heisenberg groups.
problem None explicitly stated; focus on method.
method Computes minimal polynomials for generalized Heisenberg groups.
result Explicit minimal polynomials for generalized Heisenberg groups.
Minimal surfaces in 8D smooth and nondegenerate.
problem Generic regularity of minimal hypersurfaces in 8D.
method Analysis of C∞-generic metrics. result All minimal hypersurfaces are smooth and nondegenerate.
Sharpness minimization algorithms don't solely improve generalization.
problem Why do overparameterized neural networks generalize?
method Theoretical and empirical investigation of two-layer ReLU networks.
result Sharpness minimization algorithms do not always lead to better generalization.
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 study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.
problem Finding regularity of minimal hypersurfaces in Riemannian manifolds.
method Estimate for a one-parameter min-max minimal hypersurface.
result Generic regularity of minimal hypersurfaces in 8-dimensional Riemannian manifolds with positive Ricci curvature.
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…
Observing a linear superposition principle, a family of new minimal hypersurfaces in Euclidean space is found, as well as that linear combinations of generalized helicoids induce new algebraic minimal cones of arbitrarily high degree.
The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.
problem The behavior of singular minimal hypersurfaces in dimensions 7 and above.
method Analyzes the local behavior of minimal hypersurfaces under perturbations and convergence of families of hypersurfaces.
result Existence and smoothness of nearby minimal hypersurfaces under perturbations, uniqueness of homological minimization, and existence of Jacobi fields.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…
Generalizes Thomas-Yau theorem for special and minimal Lagrangians.
problem Proving uniqueness of special Lagrangians and minimal Lagrangians.
method Hamiltonian perturbations using Imagi, Joyce, and Oliveira dos Santos method.
result Generalized uniqueness theorem for special and minimal Lagrangians.
Proves minimality of tensor varieties, generalizing previous results.
problem Finding minimality conditions for tensor varieties.
method Using Lawlor's curvature criterion and deriving Lawlor's ODE.
result Proves minimality of a class of tensor varieties except for one case.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.
problem Finding smooth minimizing hypersurfaces in high dimensions.
method Analyzing the Plateau problem and area minimization in integral homology.
result Smooth minimizing hypersurfaces are generic in 11D, with singularities in higher dimensions.
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.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
Minimal hypersurfaces in spheres generated by isoparametric foliations are found.
problem Existence of minimal hypersurfaces in spheres generated by isoparametric foliations.
method Generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, reducing the minimal surface equation to an ordinary differential equation.
result Closed embedded minimal hypersurfaces of topological type S1imesM are found for any isoparametric hypersurface M⊂Sn. Regret minimization is a powerful tool for solving large-scale problems; it was recently used in breakthrough results for large-scale extensive-form game solving. This was achieved by composing simplex regret minimizers into an overall regret-minimization framework for extensive-form game strategy spaces. In this paper…
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
Generic smooth minimal hypersurfaces exist in 8D manifolds.
problem Existence of smooth minimal hypersurfaces in high-dimensional manifolds.
method Global perturbation argument and a novel geometric invariant.
result Generic metrics on 8D manifolds admit smooth minimal hypersurfaces.
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…
Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.
problem Understanding scarring behavior of minimal hypersurfaces along stable ones.
method Analyzing a generic metric on a manifold to show scarring of minimal hypersurfaces.
result Closed, embedded minimal hypersurfaces scarring along stable ones, with diverging area and Morse index.
Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.
problem Uniqueness of generalized Gauss maps for minimal surfaces with shared hypersurfaces in projective varieties.
method Analysis of minimal surfaces in Rn+1 with inverse images of hypersurfaces in a projective subvariety. result Generalization and improvement of previous results on the uniqueness of generalized Gauss maps.
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.
We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod n homology. 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.
Paper solves the minimal generating set problem for singular Reidemeister moves.
problem Determine minimal generating sets of oriented singular Reidemeister moves.
method Introduced new invariant for singular links to detect type IV moves and provide obstructions.
result Proved exactly 96 distinct inclusion-minimal generating sets for singular moves.
It is shown that if a minimal ruled surface admits a Kähler Yamabe minimizer, then this metric must be generalized Kähler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.
Minimal hypersurfaces scarring along a fixed one in certain manifolds.
problem Scarring of minimal hypersurfaces in specific manifolds.
method Generic scarring phenomenon for minimal hypersurfaces in thick-at-infinity manifolds with thin foliation.
result Existence of sequences of minimal hypersurfaces scarring along a fixed one, with diverging area and renormalized convergence to the fixed hypersurface.
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.
We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.
Minimal generating sets found for Kim-Manturov groups.
problem Understanding the structure of groups related to surface triangulations.
method Provided minimal generating sets and determined abelianizations.
result Minimal generating sets and abelianization results for the groups.
Paper proves existence of minimal doublings on surfaces with specific properties.
problem Existence of minimal doublings on surfaces with given properties.
method Variational approach to finding nondegenerate critical points of a Coulomb-type energy.
result Proves existence of minimal doublings for surfaces of index one in a generic 3-manifold.
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…
Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such pa…
Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.
problem Singularities of area minimizing hypersurfaces.
method Perturbation of singularities.
result Singularities can be perturbed away in dimensions 9 and 10.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
In this paper, we show that a closed manifold Mn+1(n≥7) endowed with a C∞-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for 2≤n≤6, our argument also implies the denseness of the minimal hypersurfaces realizing min-m…
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…
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
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 harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature ϰ satisfy the inequality K2−ϰ2>0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …
Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.
problem Determining minimal surfaces from boundary data.
method Developed a semiclassical nonlinear calculus for complex geometric optics solutions.
result Minimal surfaces can be recovered from the Dirichlet-to-Neumann map under certain conditions.
The Weierstrass representation for minimal surfaces in R3 provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…