New Calabi-Yau metrics converge polynomially to Calabi model space.
problem Finding complete Calabi-Yau metrics with polynomial convergence rate.
method Defined new metrics on Calabi-Yau complements with ample normal bundles.
result Uniqueness of these metrics within a cohomology class.
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Compactifies Calabi-Yau to weak Fano manifolds.
problem Compactifying Calabi-Yau manifolds to weak Fano manifolds.
method Generalized Tian-Yau construction and asymptotically Calabi metrics.
result Calabi-Yau structure arises from compactification.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and L∞-estimates for Kähler potentials. result Uniform Skoda type estimate and L∞-estimate for Calabi-Yau Kähler potentials proved. Classifies Calabi hypersurfaces with parallel Fubini-Pick form.
problem Classifying Calabi hypersurfaces with specific geometric properties.
method Classification based on parallel Fubini-Pick form and Levi-Civita connection.
result Classification of 2 and 3-dimensional Calabi hypersurfaces.
For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
The abstract discusses how generalized Calabi-Gray manifolds help solve non-Kähler geometry questions.
problem Non-Kähler complex manifolds with explicit geometry.
method Demonstrates the use of generalized Calabi-Gray manifolds to address non-Kähler geometry questions.
result Generalized Calabi-Gray manifolds provide a framework to answer specific non-Kähler geometry problems.
Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on…
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T4. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group Z. In this paper, we also produce the first examples of simply con…
Holomorphic connections on Calabi-Yau manifolds are flat.
problem Existence of holomorphic connections on Calabi-Yau manifolds.
method Proving the existence of flat holomorphic connections for holomorphic vector bundles.
result Holomorphic vector bundles over compact Kähler Calabi-Yau manifolds admit flat holomorphic connections.
Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.
Sharp diameter bounds for Calabi-Yau degenerations proved.
problem Bounding the diameter of Calabi-Yau metrics during degeneration.
method Sharp upper and lower bounds derived for Ricci-flat Kahler metrics.
result Conjecture confirmed by obtaining precise diameter bounds.
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.
This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.
In this paper, we obtain several a-priori estimates for the Calabi flow on projective bundles admitting the generalized Calabi constructions.
The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths.
method Introducing combinatorial Calabi flows and proving their long time existence and global convergence.
result Proves the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary.
We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural p…
Study on Calabi-Yau metrics and their degenerations.
problem Understanding degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds.
method Survey of recent results on questions about degenerations.
result Survey of recent results on degenerations of metrics.
This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
New complete Calabi-Yau metrics found in complex space.
problem Finding metrics on complex spaces with specific conditions.
method Generalized Calabi ansatz, non-archimedean Monge-Ampère equation.
result Complete Calabi-Yau metrics constructed in Fano manifolds.
Calabi flow extended with bounded curvature integrals.
problem Extending Calabi flow on compact Kähler manifolds.
method Using bounded Lp scalar curvature integrals. result Calabi flow can be extended under certain curvature conditions.
Study noncommutative deformations of Calabi-Yau threefolds.
problem Understanding the geometry of Calabi-Yau threefolds under noncommutative deformations.
method Analyzing the influence of Poisson structures on quantum moduli spaces.
result The choice of Poisson structure significantly affects the geometry of quantum moduli spaces.
Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.
problem Constructing special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds.
method Constructs special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces.
result Special Lagrangian submanifolds shrink to 1-dimensional graphs in the base as the 3-folds collapse.
Study on collapsing Calabi-Yau manifolds and their metrics.
problem Understanding degenerations of Calabi-Yau manifolds with Ricci-flat Kahler metrics.
method Survey of recent developments, focusing on volume collapsing metrics.
result New insights into the behavior of Calabi-Yau manifolds under volume collapse.
A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact L…
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
Calabi flow works well with bounded curvature on compact manifolds.
problem Stability of extremal Kähler metrics under the Calabi flow.
method Extending the Calabi flow with Lp scalar curvature bounds. result Calabi flow converges exponentially to extremal Kähler metrics.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. Proves certain Calabi-Yau varieties are projective.
problem Compact Calabi-Yau varieties with isolated singularities are not always projective.
method Analysis and Ohsawa's degenerate spectral sequence in higher dimensions.
result Proves compact Calabi-Yau varieties with certain isolated singularities are projective.
Study shows Calabi-Yau metrics converge to a specific form under certain conditions.
problem Degeneration of Calabi-Yau metrics and their limits.
method Optimal transport problem and minimisation of Kontorovich functional.
result Limit data of Calabi-Yau metrics can be encoded into a unique minimiser.
The paper proves a numerical condition for solving complex Hessian quotient equations with Calabi symmetry.
problem Solvability of complex Hessian quotient equations with specific symmetry.
method Proving a numerical condition and proposing a conjecture on existence of k-subharmonic representatives. result Numerical condition ensures solvability of complex Hessian quotient equations.
Study the limit of Calabi-Yau metrics with degenerate skeletons.
problem Understanding the behavior of Calabi-Yau metrics with degenerate skeletons.
method Using polarised degenerations and optimal transport problems.
result Describe the limiting behaviour of the Calabi-Yau potential.
Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.
problem Characterizing Calabi-Yau locally conformally Kähler manifolds.
method Analyzing properties of Calabi-Yau locally conformally Kähler manifolds, proving they are Vaisman.
result Calabi-Yau locally conformally Kähler manifolds are Vaisman.
Estimates symplectic Calabi-Yau equation using Cheng-Yau method.
problem Estimating solutions to the Calabi-Yau equation on symplectic 4-manifolds.
method Applies a Cheng-Yau type estimate in the symplectic setting.
result Proves an a priori estimate for the symplectic Calabi-Yau equation.
Introduces new limit spaces for degenerating Calabi-Yau families.
problem Understanding degenerating Calabi-Yau families and their limit structures.
method Introduces galaxy spaces as dense subspace of infinite open Calabi-Yau varieties.
result Galaxy spaces are projective limits of toroidal compactifications.
The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
problem Classifying diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
method Using differential-geometric gluing method and classifications of simply-connected 6-manifolds.
result Any two doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.
We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat C3 or a Calabi-Yau structure does. The underlying almost complex structure may not be integrable. Such SU(3)-structures ar…
We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.