Survey new geometric inequalities for the Monge-Ampere equation.
problem Geometric inequalities for the Monge-Ampere equation.
method Survey of new geometric inequalities and estimates for the Monge-Ampere equation.
result New geometric inequalities and estimates for the Monge-Ampere equation.
Study limits of quasi-local angular momentum at infinity of gravitating systems.
problem Understanding limits of quasi-local angular momentum at infinity of gravitating systems.
method Based on optimal isometric embedding and quasilocal mass theory, the study defines and analyzes the limits of quasi-local angular momentum at spatial and null infinity.
result Limits of quasi-local angular momentum are discussed at spatial and null infinity of an isolated gravitating system.
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
Study of quasilocal mass using isometric embedding in various spacetimes.
problem Understanding quasilocal mass in different spacetimes.
method Application of isometric embedding theory to quasilocal mass.
result Recent progress in quasilocal mass calculations with specific spacetimes.
In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …
We discuss the concepts of energy and mass in relativity. On a finitely extended spatial region, they lead to the notion of quasilocal energy/mass for the boundary 2-surface in spacetime. A new definition was found in [27] that satisfies the positivity, rigidity, and asymptotics properties. The definition makes use of …
These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall i…
New method resolves ambiguity in measuring black hole merger angular momentum.
problem Ambiguity in measuring angular momentum during black hole mergers.
method Quasilocal mass and optimal isometric embedding theory.
result New definition of angular momentum free of supertranslation ambiguity.
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
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.
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.
New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.
problem Donaldson-Uhlenbeck-Yau theorem implications
method Geodesic rays of Hermitian metrics
result New proofs of the theorem
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. 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 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.
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.
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…
Reviewed and extended Wang-Yau quasi-local mass definitions.
problem Clarifying and extending definitions of quasi-local mass.
method Reviewed existing definitions and extended positivity proof.
result Extended positivity proof for Wang-Yau quasi-local energy.
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.
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.
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.
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.
The Fu-Yau equation is an equation introduced by J. Fu and S.T. Yau as a generalization to arbitrary dimensions of an ansatz for the Strominger system. As in the Strominger system, it depends on a slope parameter α′. The equation was solved in dimension 2 by Fu and Yau in two successive papers for α′>0, and for $…
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.
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.
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…
Sharp gradient estimates extended to surfaces with lower Ricci curvature.
problem Rigidity of Cheng-Yau gradient estimates on surfaces with lower Ricci curvature.
method Extending Cheng-Yau gradient estimates to surfaces with lower Ricci curvature bound and higher-dimensional Riemannian manifolds.
result Pointwise Cheng-Yau gradient estimates for higher-dimensional Riemannian manifolds and monotonicity formulas for positive harmonic functions.
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.
The paper explores the Thomas-Yau conjecture using holomorphic curves and Floer theory.
problem Proving the Thomas-Yau conjecture in the context of Lagrangian branes.
method Using holomorphic curves and Floer theory to construct and analyze bordism currents and moduli spaces.
result Established Floer theoretic obstructions and variational framework for finding special Lagrangians.
Study on asymptotic behavior of Taub-NUT type solitons and construction of new ALF Calabi-Yau metrics.
problem Asymptotic behavior of steady gradient Kähler-Ricci solitons of Taub-NUT type.
method Determination of asymptotic cone, special case analysis, and construction of new metrics using Tian-Yau-Hein method.
result Construction of new ALF Calabi-Yau metrics on quotients of the Taub-NUT type soliton.
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.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
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.
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.
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 …