Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
The article proves K5 and K3,3 are toroidal penny graphs.
problem Optimal sphere packing on torus.
method Analyzing connections between planar graphs, penny graphs, and toroidal penny graphs.
result K5 and K3,3 are toroidal penny graphs. In this paper we study general rotational surfaces in the 4- dimensional Euclidean space E4 and give a characterization of flat general rotation surface with pointwise 1-type Gauss map. Also, we show that a non-planar flat general rotation surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Cl…
We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.
We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus T2⊂R3 and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecu…
Simpler algorithms for morphing planar and toroidal graphs.
problem Constructing smooth transitions between isomorphic drawings of planar and toroidal graphs.
method Barycentric interpolation and scaling strategy.
result Simplified and more natural morphs with improved computational efficiency.
The paper extends knot polynomials to annular and toroidal pseudo links.
problem Analyzing pseudo links with undefined crossings.
method Introducing Kauffman bracket and Jones-type polynomials for annular and toroidal pseudo links.
result New tools for studying annular and toroidal pseudo links.
We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.
The study extends Tutte's conflict graph concept to nonplanar graphs.
problem Understanding the structure of nonplanar graphs through conflict graphs.
method Defining a signed conflict graph for maximally planar subgraphs and analyzing their balance.
result For graphs with a flat embedding, every maximal planar subgraph has unbalanced conflict graphs if and only if the graph is intrinsically linked.
The paper extends knotoid theory to annular and toroidal settings.
problem Extending knotoid theory to new geometric settings.
method Introducing new knotoid types, extending bracket polynomials.
result Universal bracket polynomials for annular and toroidal knotoids.
The paper extends knot theory to annular and toroidal pseudo knots.
problem Defining and classifying pseudo knots in annular and toroidal settings.
method Introducing pseudo knots as equivalence classes under moves, lifting to torus, and exploring inclusion relations.
result New invariants for classifying pseudo knots and links in solid and thickened torus.
Origami creates flat torus models of any size.
problem Creating flat torus models of any size.
method Explicit origami folding instructions.
result Flat torus models of any size created.
Flat torus triangulations' space is homotopy equivalent to a torus.
problem Proving homotopy equivalence of triangulations of flat tori.
method Generalization of Tutte's embedding theorem for flat tori.
result Deformation space of geodesic triangulations is homotopy equivalent to a torus.
We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-…
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere S3=R3∪{∞}. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type…
We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …
Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in C2. result Asymptotic counting result for tripods on the torus.
New proof confirms flat equilateral torus is λ1-maximal.
problem Maximizing the first eigenvalue on flat tori.
method Combining El Soufi-Ilias-Ros's method and Bryant's result.
result Positive answer to Berger's isoperimetric problem.
We extend our discrete uniformization theorems for planar, m-connected, Jordan domains [Journal für die reine und angewandte Mathematik 670 (2012), 65--92] to closed surfaces of non-positive genus.
Study shows convergence of certain metrics to flat torus.
problem Stability of metrics on three-torus with negative scalar curvature.
method Defined metrics and used Stern's inequality to show convergence.
result Subsequence of metrics converges to flat metric.
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
Consider a planar, bounded, m-connected region Ω, and let $\bordΩ$ be its boundary. Let T be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f) where S is a genus (m−1)…
Universal triangulation for flat tori with 2434 triangles.
problem Embedding flat tori isometrically in 3D space.
method Adapted Burago and Zalgaller's proof for polyhedral surfaces, combined with Zalgaller's construction.
result A universal triangulation of 2434 triangles for any flat torus.
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
Proves product metrics are Yamabe metrics under small flat torus conditions.
problem Yamabe metrics on product spaces with small flat tori.
method Extends earlier results to Type~I and Type~II Yamabe constants, Q-curvature problems, and isoperimetric-ratio type problems. result Product metrics are Yamabe metrics for sufficiently small flat tori.
Hamilton flows on Kähler manifold for which all trajectories are H-planar curves (complex analog of geodesics) are considered. These flows are called H-planar. The equation which has to obey the Hamiltonian of H-planar Hamilton flow is received and the method of finding general solution of this equation is propos…
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…
Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.
problem Deforming foliated manifolds with flat leaves while maintaining curvature bounds.
method Collapsing a manifold with a closed flat regular Riemannian foliation, keeping curvature uniformly bounded.
result For compact, simply connected manifolds, foliations are given by torus actions.
By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subco…
Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs K3,3 and K5 on the double torus. method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus. New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional cu…
Proves uniqueness and existence of toric gravitational instantons.
problem Proves uniqueness and existence of four-dimensional asymptotically flat, Ricci-flat, toric gravitational instantons.
method Adapting black hole uniqueness theorems to a harmonic map formulation of Ricci-flat metrics with torus symmetry.
result Proves that instantons are uniquely characterised by their rod structure and that for every admissible rod structure, there exists a smooth instanton.
Classifies symmetries of non-flat 3-webs around a point.
problem Understanding symmetries of non-flat 3-webs.
method Classification and construction methods for symmetries.
result Classification of symmetries for non-flat 3-webs.
Analytic non-planar p-elasticae are shown to be 3D.
problem Classifying non-planar p-elasticae. method Analyticity and structure results for p-elasticae in Rn. result Every non-planar p-elastica is analytic and three-dimensional. Study SKT and CYT manifolds with parallel Bismut torsion.
problem Characterize and construct compact complex manifolds with specific geometric properties.
method Characterization via universal cover, construction using mapping torus, investigation of generalized Kaehler structures.
result Existence of non-Bismut flat examples and characterization of universal covers.
Functors from web categories differ despite similar definitions.
problem Distinguishing between combinatorial and gauge-theoretic evaluations of webs.
method Exhibited a counterexample showing J♯ restricted to planar webs is not J♭. result Restriction of J♯ to planar webs is distinct from J♭. We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti…
Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
In this paper, we consider discrete groups in PGLd(R) acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for CAT(0) spaces.