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.
Generalizes twistor lines for complex tori, introducing new non-compact curves.
problem Understanding the structure of complex tori through twistor lines.
method Introducing and studying two new types of non-compact analytic curves in the period domain of complex tori.
result Analytic properties of compactifications of curves, preservation of cohomology classes, and twistor path connectivity.
We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space.
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…
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.
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.
New method detects essential tori in mixed singularity links.
problem Detecting essential tori in mixed singularity link complements.
method Analyzing properties of defining mixed polynomials.
result Explicit criteria for essential tori existence.
Study moduli spaces of flat tori using analytical and cohomological methods.
problem Understanding moduli spaces of flat tori with conical singularities and holonomy.
method Analytical and cohomological methods, including surgeries for flat surfaces.
result Explicit constructions and geometric results for moduli spaces of flat tori.
Researchers compute zeta-determinants and analytic torsion for metric mapping tori.
problem Computing zeta-determinants and analytic torsion for metric mapping tori.
method Using the BFK-gluing formula for zeta-determinants.
result Computed zeta-determinants and analytic torsion for metric mapping tori.
Researchers found parallel mean curvature tori in complex projective and hyperbolic planes.
problem Finding tori with parallel mean curvature vectors.
method Explicit determination of tori in complex projective and hyperbolic planes.
result Explicit determination of tori with parallel mean curvature vectors in both complex projective and hyperbolic planes.
Classifies all orthogonal complex structures on a flat 6-torus.
problem Classify orthogonal complex structures on a flat 6-torus.
method Analyzes flat tori and identifies all orthogonal complex structures.
result All orthogonal complex structures on a flat 6-torus are either complex tori or BSV-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.
Flat tori found non-isometric pairs with identical Laplace eigenvalues.
problem Finding the lowest dimension for isospectral non-isometric flat tori.
method Analytic, geometric, and number theoretic approaches.
result Schiemann resolved the isospectral problem for flat tori in the 1990s.
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. Twistor lines connect complex tori in their period domain.
problem Connecting complex tori in their period domain.
method Analyzing twistor lines in the period domain of complex tori.
result Periods of complex tori can be joined by generic chains of twistor lines.
Study on complex tori foliations and flat geometries.
problem Understanding turbulent foliations on compact complex tori.
method Defined and analyzed smooth turbulent foliations on compact complex tori.
result All transversely holomorphic Cartan geometries are flat.
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.
New method bounds curvature of Calabi flow on complex tori.
problem Smoothness of Calabi flow on complex tori.
method Developed a new method to obtain explicit curvature bounds.
result Calabi flow starting from weak Kähler metric becomes smooth immediately on n=2 complex tori. Lower bound found for energy on specific Lagrangian tori in complex projective space.
problem Finding a lower bound for the energy functional on Lagrangian tori in CP2. method Analyzing the energy functional on a family of Hamiltonian minimal Lagrangian tori.
result Proved that the energy of certain Hamiltonian minimal Lagrangian tori is strictly larger than the Clifford torus.
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. Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
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.
The paper studies energy functionals for Lagrangian tori in complex projective space.
problem Investigating energy functionals for Lagrangian tori in complex projective space.
method Introducing an energy functional based on the potential of associated Schrödinger operators and studying its behavior on specific families of tori.
result Proposes that the minimum of the energy functional is achieved by the Clifford torus.
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.
The paper develops flows for tori and spheres, addressing complex geometries.
problem Learning flows on tori and spheres for complex geometries.
method Recursive flows starting from circles, intervals, or spheres.
result Expressive and numerically stable flows on tori and spheres.
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.
Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.
problem Analyzing anomaly terms in spaces with conical singularities.
method Proof of vanishing anomaly term for smooth conical spaces; demonstration of non-triviality for cones over tori.
result Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.
The paper studies exact triangles in stable vector bundles on tori.
problem Understanding exact triangles in stable vector bundles on tori.
method Geometric interpretation via Fukaya category and homological mirror symmetry.
result Geometric interpretation of exact triangles in terms of Fukaya category.
The paper proves existence and behavior of Lagrangian tori in complex projective plane.
problem Existence and behavior of Lagrangian tori in complex projective plane.
method Lagrangian mean curvature flow with surgery.
result Existence of monotone Lagrangian tori under Lagrangian mean curvature flow in complex projective plane.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
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.
Holomorphic structures on tori are mostly invariant under translations.
problem Understanding symmetries of geometric structures on complex tori.
method Analyzing holomorphic locally homogeneous geometric structures on complex tori.
result Holomorphic structures on tori are translation invariant.
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…
Study circle patterns on tori, linking symplectic forms and homeomorphisms.
problem Understanding circle patterns on tori and their symplectic properties.
method Investigates the space of circle patterns on closed tori with complex projective structures, embedding it into Teichmüller spaces and analyzing symplectic forms.
result Non-degeneracy of the pulled-back Weil-Petersson symplectic form and homeomorphism between circle patterns and Teichmüller spaces.
The complexity of horizontality in twistor spaces on tori is infinite.
problem Complexity of horizontality in twistor spaces on tori.
method Analyzing the complexity of horizontality in the twistor space associated with an oriented vector bundle over a torus.
result The complexity of horizontality in the twistor space is expressed by a dense subset of S2 when it is infinite. 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…
Constructs maps from field theories to complexified K-theory and elliptic cohomology.
problem Identifying geometric models for Chern characters in supersymmetric field theories.
method Higher-dimensional generalization of Fei Han's method, involving super moduli spaces and derived geometry.
result Provides evidence for the Stolz--Teichner program and geometric models for Chern characters.
We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to any symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure.
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…
The paper examines Hamiltonian stability of Lagrangian tori in complex hyperbolic spaces.
problem Investigating Hamiltonian stability of Lagrangian tori in complex hyperbolic spaces.
method Standard Hamiltonian Tn-action on CHn; proving stability and rigidity results. result Existence of infinitely many H-unstable Tn-orbits when n≥3. I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…
It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in the complex projective plane may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spect…
Affirmative answer to flat holomorphic Cartan geometries on complex tori.
problem Whether all flat holomorphic Cartan geometries on complex tori are translation invariant.
method Using complex affine Lie groups, we show that all holomorphic Cartan geometries on complex tori are translation invariant.
result Holomorphic Cartan geometries on complex tori are translation invariant.
We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.
This is an expository article which describes one approach to the construction and classification of harmonic tori "of finite type", namely, via their ring of polynomial Killing fields. To keep the discussion focussed, the first section is devoted entirely to non-conformal harmonic tori in the 2-sphere. The second sect…
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Constructs exotic Lagrangian tori in Grassmannians using cluster algebra.
problem Finding non-displaceable and non-isotopic Lagrangian tori in Grassmannians.
method Iterative construction based on cluster algebra structure of a mirror Landau-Ginzburg model.
result Examples of exotic Lagrangian tori that support nonzero objects in different summands of the Fukaya category.
Floer homology reveals incompressible tori in homology spheres.
problem Existence of incompressible tori in homology spheres.
method Computation of bordered Floer homology.
result Homology spheres without incompressible tori have specific properties.