Paper shows convergence types match for almost minimal sets.
problem Matching convergence types for almost minimal sets.
method Hausdorff and varifold convergence comparison on almost minimal sets.
result Hausdorff and varifold convergence coincide on almost minimal sets.
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.
Study on biharmonic almost complex structures on compact manifolds.
problem Existence and regularity of biharmonic almost complex structures.
method Analyzes biharmonic almost complex structures on compact almost Hermitian manifolds, focusing on dimension four.
result Existence of energy-minimizing biharmonic almost complex structures for various topologies and homotopy classes.
Constructs approximating functions for almost minimal 2D currents.
problem Approximating carefully integral 2D currents with small cylindrical excess.
method Constructs Lipschitz Q-valued functions. result Proves discreteness of singular set for specific 2D currents.
Study analyzes behavior of almost minimal 2D currents at singular points.
problem Understanding the behavior of almost minimal 2D currents at singular points.
method Analyzes asymptotic behavior of currents in Riemannian manifolds, semicalibrated, and spherical cross sections of 3D area minimizing cones.
result Shows the discreteness of the singular set for the analyzed classes of currents.
Constructs a branched center manifold for almost minimal 2D currents.
problem Understanding the structure of singular points in almost minimal 2D currents.
method Constructs a branched center manifold near singular points.
result Shows the discreteness of singular set for specific current classes.
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
The study classifies weakly almost Fuchsian manifolds and proves geometric properties.
problem Classifying and understanding weakly almost Fuchsian manifolds.
method Geometric analysis and compactification techniques.
result Uniform upper bounds on volume and Hausdorff dimension for limit sets.
The paper proves unique tangent cones for 2D almost minimizing currents.
problem Proving uniqueness of tangent cones for 2D almost minimizing currents.
method Unified approach based on White's theorem for area minimizing currents, with modifications.
result Tangent cones are everywhere unique for 2D almost minimizing currents.
The paper analyzes minimal G-structures induced by the Lee form.
problem Analyzing minimal G-structures for specific classes of manifolds.
method Computing conditions of minimality for Gray-Hervella classes and comparing with the Lee form.
result The Lee form represents minimal G-structures in certain classes of manifolds.
The study classifies natural almost Hermitian structures on specific Lie groups.
problem Classifying natural almost Hermitian structures on conformally foliated Lie groups.
method Examining 4-dimensional Riemannian Lie groups with a 2-dimensional conformal foliation and minimal leaves, constructing new examples of multi-dimensional structures.
result Constructing several new multi-dimensional examples of almost Kähler, integrable, and Kähler structures.
Study shows almost minimizing rectifiable chains in Hilbert space have regular points dense in their support.
problem Understanding the regularity of almost minimizing rectifiable chains in infinite dimensional spaces.
method Adapted Reifenberg's epiperimetric inequality and computations by Preiss to infinite dimensional space.
result The set of regular points is dense in the support of almost mass minimizing rectifiable G chains. Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
The paper proves smoothness of almost-minimizers' boundaries near the free boundary.
problem Minimizing degenerate area functionals with weighted boundary conditions.
method Epsilon-regularity theorem applied to almost-minimizers.
result Almost-minimizers' boundaries are C1,γ0-smooth, orthogonal to the boundary Ω. Study soap films hanging from frames, proving surface limits and curvature conditions.
problem Understanding soap films with gravity and minimal surfaces.
method Compactness theorem for surfaces with vanishing mean curvature and fixed/converging boundaries.
result Minimal surfaces represent all possible limits of almost-minimal surfaces.
Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.
problem Analyzing H-minimal Legendrian surfaces in Heisenberg group.
method Proved an almost monotonicity formula.
result Deduced a Bernstein-Liouville type theorem.
The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.
problem Classifying natural almost Hermitian structures on Lie groups with minimal conformal leaves.
method Analyzing Lie groups with a 2-dimensional conformal foliation and classifying structures based on Lie algebra properties.
result 16 multi-dimensional almost Kähler families, 18 integrable families, and 11 Kähler families were constructed.
New examples show flat singular sets can be arbitrarily complex.
problem Understanding the structure of singular sets in almost-minimizing currents.
method Construction of specific examples of area almost-minimizing currents.
result Flat singular sets can contain any closed empty interior subset of a plane.
We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
Minimal surfaces can have tiny undulations.
problem Existence of minimal surfaces with controlled geometry.
method Construction of almost flat minimal graphs with micro-oscillations.
result Existence of minimal graphs with prescribed intersection geometry.
Boundary bubbles form in almost minimal cylinders under specific conditions.
problem Analyzing the asymptotic behavior of solutions to the Teichmüller harmonic map flow.
method Analyzes the Teichmüller harmonic map flow and almost minimal cylinders under Plateau-boundary conditions.
result Boundary bubbles form and are branched minimal immersions under Douglas' separation condition.
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
New epiperimetric inequality for cones with improved regularity results.
problem Regularity of almost area-minimizing currents at singular points.
method Flowing in radial direction for cones with isolated singularities.
result New ε-regularity result for almost area-minimizing currents.
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…
We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.
Study horospheres in hyperbolic 3-manifolds with degenerate ends.
problem Understanding geodesics in degenerate hyperbolic 3-manifolds.
method Analyze almost minimizing geodesics to explore horospheres.
result Identify geodesics passing through thin parts of the manifold.
Paper classifies CFK∞ type of almost L-space knots.
problem Understanding almost L-space knots and their properties.
method Utilizes Heegaard Floer homology and knot Floer homology.
result Classifies CFK∞ type of almost L-space knots. For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…
New 5-manifold found with zero Ricci curvature.
problem Finding almost Ricci-flat manifolds with specific properties.
method Kummer-type constructions using Riemannian metrics.
result Constructed a simply connected, nonspin 5-manifold with zero Ricci curvature.
Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one ca…
In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is "almost graph…
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class W1,p, 1<p<∞. If f almost minimizes its p Dirichlet energy then f is Hölder continuous. If p=2 and f is squeeze and squash stationary then f is in VMO.
In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers …
The paper extends toric variety correspondence to 4D almost complex torus manifolds.
problem Extending toric variety correspondence to 4D almost complex torus manifolds.
method Associate combinatorial objects (families of multi-fans and graphs) to 4D almost complex torus manifolds and find conditions for their equivalence.
result Minimal models and operations for combinatorial objects, and equivalence between 4D complex torus manifolds and their minimal models.
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…
Paper proves minimality of stable submanifolds in flat neighbourhoods.
problem Proving the uniqueness of minimizers of elliptic functionals in flat neighbourhoods.
method Introducing a penalized minimization problem to exploit almost minimizers' regularity.
result Quantitative estimates of minimizers in flat neighbourhoods.
Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.
problem Mean curvature flow of totally real submanifolds.
method Quantitative analysis of almost minimal submanifolds.
result Established convergence result for mean curvature flow.
The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.
problem Investigating the geometry of gradient almost Yamabe solitons in warped product manifolds.
method Presenting geometric rigidity results, investigating existence conditions, and classifying specific solitons.
result Classification of rotational gradient almost Yamabe solitons in RimesfRn. The study shows conditions for almost complex structures on rational homology spheres and limits on their Betti numbers.
problem Conditions for almost complex structures on rational homology spheres.
method Analyzes rational homology spheres and provides examples of manifolds with almost complex structures.
result Closed almost complex manifolds with sum of Betti numbers three have dimensions that are powers of two.
An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…
The paper classifies almost Yamabe solitons on hypersurfaces and submanifolds in Euclidean spaces.
problem Classifying almost Yamabe solitons on various geometric structures.
method Analyzing hypersurfaces and submanifolds with position and concurrent vector fields.
result Complete classification of almost Yamabe solitons on hypersurfaces and submanifolds.
An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by…
Paper explores cohomology classes on non-compact almost Kähler manifolds.
problem Understanding cohomology classes induced by symplectic forms.
method Provides criteria for non-trivial classes in Lp cohomology. result Symplectic forms induce non-trivial classes in Lp cohomology. Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to character…
The paper studies ∂ˉ-harmonic maps between almost Hermitian manifolds and derives their Euler-Lagrange equation.
problem Energy minimization of maps between almost Hermitian manifolds.
method Derives the ∂ˉ-harmonic map equation and proves analogous results to harmonic maps. result The ∂ˉ-harmonic map equation coincides with the harmonic map equation up to first order terms. The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.