Study thermodynamic framework for Monge-Ampère equations on real tori.
problem Monge-Ampère equations on real tori.
method Thermodynamic framework, point processes, convergence in law.
result Convergence in law of point processes associated with Monge-Ampère equations.
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.
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.
In this paper we show that all totally real superconformal minimal tori in CP2 correspond with doubly-periodic finite gap solutions of the Tzitzeica equation ωzz=e−2ω−eω Using the results on the Tzitzeica equation in integrable system theory, we describe explicitly all these tori by Prym-theta…
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.
Real Lagrangian tori in S2imesS2 are Hamiltonian isotopic to the Clifford torus.
problem Unknottedness of real Lagrangian tori in S2imesS2. method Neck-stretching argument, Gromov's foliation theorem, Cieliebak-Schwingenheuer criterion.
result Real Lagrangian tori in S2imesS2 are Hamiltonian isotopic to the Clifford torus. We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curv…
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.
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.
The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in S3 result when these spectral curves satisfy periodicity conditions. We prove that the spectra…
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.
The study links Ricci curvature and convexity in complex tori.
problem Characterizing Ricci curvature signs in toric manifolds.
method Characterization through convexity of volume functional.
result Relationships between Ricci curvature, volume, submanifolds, and pluri-subharmonic functions.
Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisa…
We prove existence results that give information about the space of minimal immersions of 2-tori into S3. More specifically, we show that \begin{enumerate} \item For every positive integer n, there are countably many real n-dimensional families of minimally immersed 2-tori in S3. Every linearly ful…
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.
New length functions on mapping class groups linked to simplicial volumes of mapping tori.
problem Understanding the relationship between mapping class groups and simplicial volumes of mapping tori.
method Introducing filling volumes as length functions and proving their properties.
result Real filling volumes equal the simplicial volume of mapping tori, while integral filling volumes are not smaller than the stable integral simplicial volume.
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is well kn…
A Hamiltonian stationary Lagrangian submanifold of a Kaehler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kaehler manifold of real dimension four that guarantees the existence of a family of small Hamiltonian stationar…
Study of complex tori using twistor triangles and algebraic representations.
problem Understanding the geometry of complex tori through twistor triangles.
method Using representation theory of algebras to analyze the period domain of complex tori.
result Introduced pseudometric invariants to distinguish triangles up to G-equivalence. Study minimal surfaces in 4D, find specific tori with total curvature -8π.
problem Find complete proper non-holomorphic minimal tori in R^4 with total curvature -8π.
method Use tools like Gauss maps and link/braid/writhe at infinity. Translate problem into a system of equations involving Weierstrass function. Explicitly solve for rectangular and square tori.
result Explicit solutions for minimal tori in R^4, including generalization of Chen-Gackstetter torus in R^3.
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…
The paper finds new constrained Willmore minimizers for non-rectangular tori.
problem Finding constrained Willmore minimizers for non-rectangular tori.
method Analyzing immersed tori in 3-space to minimize Willmore energy.
result The candidates constructed in previous work are constrained Willmore minimizers in certain non-rectangular conformal classes.
Classifies real tight contact structures on lens spaces and solid tori.
problem Classifying real tight contact structures on specific 3-manifolds.
method Equivariant contact isotopy, real open book decompositions, and isolated real algebraic surface singularities.
result Unique real tight structures on S3 and RP3, at most one on L(p,±1), and bounds on the count. We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds X of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of X and holomorphic vector bundles with compatible unitary conn…
The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.
problem Understanding the Gromov-Hausdorff limits of tori with Ricci conditions.
method Constructing metrics on Rn and analyzing their limits. result The Gromov-Hausdorff limit of tori with Ricci bounds is not always a topological manifold.
Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on S1, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space o…
We consider quotients of spheres by linear actions of real tori. To each quotient we associate a matroid built out of a diagonalization of the torus action. We find the integral homology groups of the resulting quotient spaces in terms of the Tutte polynomial of the matroid. We also find the homotopy type and homology …
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.
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor …
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.
We describe a new approach to the study of the set of all simple geodesics on a hyperbolic punctured torus. We introduce a valuation on the first integral homology group of the torus. This valuation associates to each homology class the length of the unique simple geodesic in it. We show that this valuation extends to …
This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of K…
Compactifies moduli of abelian varieties and curves by attaching flat tori.
problem Compactifying moduli spaces of abelian varieties and curves.
method Explicitly attaching moduli of flat tori to abelian varieties and curves.
result Explicit determination of Gromov-Hausdorff limits of abelian varieties and curves.
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.
Study noncommutative coverings of irrational quantum tori.
problem Characterize noncommutative coverings of irrational quantum tori.
method Developed a framework for noncommutative coverings and studied irrational quantum tori.
result Characterized all connected noncommutative coverings of irrational quantum tori.
Let K be the space of properly embedded minimal tori in quotients of R3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the ex…
Study of square-tiled tori and their SL(2,Z) orbits.
problem Counting and classifying square-tiled tori.
method Natural parametrization and SL(2,Z) action.
result Exact size of every SL(2,Z) orbit.
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in Cm for m>n≥2. result At least two exact isotropic n-tori in Cm are not Hamiltonian isotopic. Constructs Riemannian foliations with exotic tori as leaves.
problem Creating Riemannian foliations with exotic tori.
method Smooth fiber bundles with exotic tori fibers and finite abelian fundamental group total space.
result Examples of Riemannian foliations with exotic tori leaves and finite abelian fundamental group total space.
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.
No complex analytic tori in complex deformations of Kummer varieties.
problem Existence of complex analytic tori in Kummer varieties.
method Proving non-existence through complex deformation analysis.
result Generic complex deformations of Kummer varieties contain no complex analytic tori.
Stable 2-lobed Delaunay tori found in 3-sphere.
problem Stability of 2-lobed Delaunay tori in the 3-sphere.
method Constrained Willmore surfaces in the 3-sphere.
result 2-lobed Delaunay tori are stable.
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.