New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
New discrete models for constant mean curvature surfaces and tori.
problem Creating discrete models for constant mean curvature surfaces and tori.
method Integrable theory of discrete polarised curves and Darboux transforms.
result Closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.
problem Proving an asymptotic expansion for spectral zeta functions on discrete tori.
method Inspired by Friedli and Karlsson's work, the authors derive an asymptotic expansion for the spectral zeta function on discrete tori.
result Similar asymptotic expansions hold for m=2 and higher dimensions, equivalent to the Epstein-Riemann conjecture.
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
We construct examples of C∞ smooth submanifolds in Cn and Rn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…
Flat minimal tori counterexamples refute Lu's second-gap conjecture.
problem Lu's second-gap conjecture about minimal surfaces in higher codimensions.
method Constructing closed embedded counterexamples for minimal surfaces.
result Constant values of S+λ2 realized by flat minimal tori are dense in (2,3), refuting the conjecture. We found unique tori with same curvatures using isometric transformations.
problem Determining if metric and mean curvature uniquely define a torus.
method Constructed Bonnet pairs of tori using isothermic surfaces and conformal transformations.
result Explicit construction of compact Bonnet pairs of tori.
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
Study on discrete surfaces with constant principal curvature for nanocarbon applications.
problem Understanding discrete geometry properties of nanocarbon materials.
method Developed discrete surface theory on 3-ary oriented trees, defined discrete principal directions, constructed examples of discrete CPC surfaces.
result Construction of discrete constant principal curvature surfaces, including discrete CPC tori.
Paper extends trigonometric summation formula with weights.
problem Trigonometric summation formula by Grigor'yan, Lin and Yau.
method Weighted trigonometric summation formula derivation.
result Extension of trigonometric summation formula.
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of L⊗k, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let Γ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of …
We consider the inverse problem of reconstructing the posterior measure over the trajec- tories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using vari…
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each…
This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
problem Classifying minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
method General construction of homogeneous minimal flat n-tori in spheres, detailed investigations of shortest vectors in lattices.
result There exists a 2-parameter family of non-congruent λ1-minimal flat 4-tori.
Characterizes conformal classes of tori using differential geometry.
problem Classifying conformal classes of tori in complex dimension 1.
method Basic differential geometry methods, contrasting with Hopf tori.
result Complete characterization of conformal classes of product and standard flat tori.
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
The paper confirms conjectures about the topology of triangulated polyhedra and geodesic triangulations on spheres.
problem Topology of spaces of convex polyhedra and Delaunay triangulations on spheres.
method Variational principles on triangulated surfaces.
result Spaces of Delaunay triangulations have the same homotopy types as their smooth counterparts on the unit 2-sphere.
We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…
New findings on isospectral tori and harmonic maps between flat tori.
problem Determining if isospectral tori are isometric using harmonic maps.
method Examined harmonic maps between flat tori, focusing on Milnor's isospectral tori.
result Milnor's isospectral tori cannot be distinguished by harmonic maps from lower-dimensional tori, but can be distinguished by higher-dimensional ones.
Constructs flows of tori in sphere perturbations for Morse homology.
problem Understanding tori in sphere perturbations.
method Constructs eternal mean curvature flows of tori.
result Constructs flows of tori in sphere perturbations.
We prove that the conformal immersions of complex two tori into S3 which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…
Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.
problem Minimal tori in ellipsoids.
method Analyzing 3D ellipsoids invariant under a 2-torus action.
result Infinitely many distinct minimal tori bifurcate from a 2-torus orbit.
Study of critical tori for mean curvature energies in Killing submersions.
problem Analyzing surface energies in Killing submersions.
method Symmetry reduction and binormal evolution of critical curves.
result Construction of vertical tori critical for mean curvature energies.
We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in Sn, as well as the explicit expressions of some of these immersions.
The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…
Isothermic tori with one planar curvature line found and characterized.
problem Classifying isothermic tori with specific curvature lines.
method Complex analytic methods and explicit theta function formulas.
result Explicit formulas for family of plane curves and their relation to hyperbolic elastica.
For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.
Study tiling spaces over irrational tori using diffeological classification.
problem Understanding the structure of tiling spaces over irrational tori.
method Diffeological classification of irrational tori and analysis of fiber bundle structures.
result Inherited diffeological equivalence of one-dimensional tiling spaces over irrational tori.
New minimal tori found in curved spaces.
problem Existence of minimal tori in curved spaces.
method Generalized Angenent's shrinking tori to minimal n-dimensional tori. result Existence of rotationally symmetric embedded f-minimal tori.
Classifies mapping tori of specific groups, generalizing known results.
problem Classifying mapping tori of specific groups.
method Using Hopf-type properties and Poincaré Duality groups.
result Generalizes and provides new proofs for fibered 3-manifolds.
Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite…
We consider smooth isotropic immersions from the 2-dimensional torus into R2n, for n≥2. When n=2 the image of such map is an immersed Lagrangian torus of R4. We prove that such isotropic immersions can be approximated by arbitrarily C0-close piecewise linear isotropic maps. If n≥3 the piece…
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
Smooth tori in S^4 are topologically unknotted.
problem Tackling the topological unknottedness of smooth tori in S^4.
method Analyzing the intersection forms and critical points of tori to prove topological unknottedness.
result Certain smooth tori in S^4 are topologically unknotted.
In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori TR2n for any n≥3. We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-to…
Counts minimal tori in Riemannian manifolds with 6 or more dimensions.
problem Counting minimal tori in Riemannian manifolds.
method Introduces a function to count minimal tori and shows invariance under metric perturbations.
result The count function is invariant under metric perturbations.
Let M be a cusped hyperbolic 3-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in PGL(2,C) (up to conjugation) is of complex dimension the number ν of cusps near the hyperbolic representation. It seems natural to ask whether some …
The paper finds non-contractible loops of Legendrian tori from knot families.
problem Computing non-contractible loops of Legendrian tori from knot families.
method Using cord algebra of knots to compute Legendrian contact homology.
result Obtained an infinite family of non-contractible loops of Legendrian tori.
The study limits the number of 2-holed tori in knot exteriors.
problem Bounding the number of 2-holed tori in knot exteriors.
method Continuing Motegi's program, the paper applies universal bounds to hyperbolic knots.
result There are at most six non-isotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
We show that for m>n≥2, there are at least two exact isotropic n-tori in Cm which are not Hamiltonian isotopic in Cm, even though they are smoothly isotopic as isotropic n-tori. We apply this discovery to obtain more distinct non-exact isotropic tori in Cm.
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.
New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.
problem Constructing new non-Kähler generalized Kähler manifolds.
method Starting from a 3-torus and a compact Kähler manifold, constructing via mapping tori.
result Obtained new non-Kähler examples and recovered known examples of generalized Kähler solvmanifolds.