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.
Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.
The paper describes a specific type of Kähler surfaces.
problem Understanding QCH Kähler surfaces with quasi-constant holomorphic sectional curvature.
method Detailed description of generalized Calabi type QCH Kähler surfaces.
result Detailed description of QCH Kähler surfaces of generalized Calabi type.
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. Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class…
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.
We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type on…
Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.
problem Analyzing the behavior of G2-structures under Laplacian and Hitchin flows.
method Investigation of Laplacian and Hitchin flows on contact Calabi-Yau 7-manifolds.
result Ancient solutions of the Laplacian flow with finite time Type I singularity and immortal solutions of the Laplacian coflow with infinite time Type IIb singularity.
Superminimal surfaces in certain Einstein manifolds have a Calabi-Yau property.
problem Characterizing superminimal surfaces in specific Einstein manifolds.
method Utilizing twistor spaces and properties of holomorphic Legendrian curves.
result Superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds can be uniformly approximated by complete superminimal surfaces.
New Spin(7) manifolds created from Calabi-Yau bundles.
problem Creating complete non-compact Spin(7) manifolds.
method Building on T2-bundles over AC Calabi-Yau manifolds. result First known examples of complete toric Spin(7) manifolds.
Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.
problem Proving a Calabi-Yau type theorem for specific manifolds.
method Existence result for bounded regions with weakly mean-concave boundary.
result Proves contractibility of certain manifolds with positive scalar curvature.
We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.
Machine learning predicts Hodge numbers of Calabi-Yau four-folds.
problem Predicting Hodge numbers of Calabi-Yau manifolds using machine learning.
method Supervised learning of Hodge numbers h^1,1 and h^3,1 for complete intersection Calabi-Yau four-folds.
result A two-branch network with feature enhancement achieves 98% precision for predicting h^3,1.
New insights into Kähler Ricci solitons and Calabi-Yau cones.
problem Understanding Kähler Ricci solitons and their relationship to Calabi-Yau cones.
method Analyzing the canonical cone of Fano manifolds and using openness of weight functions.
result The canonical cone of a product of a smooth Fano manifold and a complex projective space is a Calabi-Yau cone under certain conditions.
We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi-Yau two-folds. These give rise to eight-dimensional non-Kahler Hermitian manifolds with SU(4) structure. The eight…
The paper explores complex geometries of 3-forms on symplectic 6-manifolds.
problem Understanding the geometry of 3-forms on symplectic 6-manifolds.
method Investigation of geometries associated with 3-forms of various orbital types.
result Rich geometric structures attached to unstable 3-forms from Calabi-Yau degeneration.
We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original project…
Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kähler form ω and a holomorphic four-form Ω. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifol…
Researchers solve conformal Killing forms on Kaehler manifolds.
problem Classifying conformal Killing forms on compact Kaehler manifolds.
method Explicit determination of conformal Killing forms in middle degree.
result First examples of conformal Killing forms not from Hamiltonian 2-forms.
Study solutions and singularities of G2-structures flows on specific manifolds.
problem Investigate singularities and solutions of G2-structures flows.
method Explicit solutions and singularities of Ricci-harmonic flow, Ricci-like flows, and negative gradient flow of G2-structures on specific manifolds.
result First examples of Type I singularities of Ricci-harmonic flow and Type IIb and Type III singularities of Ricci-like flows.
In this paper we construct explicit smooth solutions to the Strominger system on generalized Calabi-Gray manifolds, which are compact non-Kähler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers.
Study explores unstable 3-forms on Calabi-Yau 3-folds.
problem Understanding degenerations of Calabi-Yau 3-folds via 3-forms.
method Investigates geometries of 3-forms on symplectic 6-manifolds.
result Unstable 3-forms reveal rich geometric properties related to SYZ conjecture.
This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…
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.
In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show that if the open manifold is Calabi-Yau, then Chi Li's compactification is projecti…
Construct M-Theory lifts of type IIA orientifolds.
problem Lift type IIA orientifolds to M-Theory.
method Construct M-Theory on twisted connected sum G2 manifolds.
result Two building blocks correspond to open and closed string sectors.
Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.
problem Geometric properties of log Calabi-Yau manifolds in two specific cases.
method Analysis of various geometric properties, focusing on Bochner principle, local triviality, polystability, and compactifiability of universal cover.
result The universal cover of X∖D is a Calabi-Yau manifold of infinite topological type when D has two components. 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.
Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.
problem Understanding the collapsed Gromov-Hausdorff limits of Calabi-Yau spaces.
method Analyzing the geometry of twisted Kähler-Einstein metrics on holomorphic fiber spaces.
result Proving the existence of conical-type singularities in the base of fiber spaces.
The curvature properties of a specific type of 6-manifold are explored.
problem Curvature identities on almost Calabi-Yau 6-manifolds with torsion.
method Analysis of the Nijenhuis tensor, torsion connection, and Ricci solitons.
result The curvature of the torsion connection is symmetric and vanishes if the norm of the torsion or scalar curvature is constant.
On a complete Calabi-Yau manifold M with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic 1-forms, which follows from a new local …
We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold i…
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing σ-extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics. result Existence of σ-extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds. A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we …
Study G_2-manifolds from glued circles and Calabi-Yau manifolds.
problem Analyse G_2-manifolds from glued circles and Calabi-Yau manifolds.
method Express nu-invariant in terms of fixpoints and gluing contributions, involving Dedekind sums and eta-function.
result Existence of compact G_2-manifolds that are not G_2-nullbordant.
New polystability theory connects Calabi-Yau varieties to gravitational instantons.
problem Understanding the structure of Calabi-Yau manifolds and their metrics.
method Introducing a new concept of poly-stability and relating it to gravitational instantons.
result Polystability is equivalent to the existence of certain gravitational instantons.
We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…
New complete hypersurfaces found in affine geometry.
problem Finding new complete hypersurfaces in affine geometry.
method Classifying special Calabi hypersurfaces and solving equations.
result Found a class of new Euclidean and Calabi complete affine hypersurfaces.
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.
In this paper we give classification of Kähler surfaces of generalized Calabi type.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
The Ricci Calabi functional is a functional on the space of Kähler metrics of Fano manifolds. Its critical points are called generalized Kähler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized Kähler Einstein metrics. As its application, we give a…
Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.
problem Ensuring the Kähler property of Calabi-Yau 3-folds under symplectic deformations.
method Established dynamical stability of Type IIA flow near stationary points.
result Stability of Type IIA flow ensures the stability of Kähler properties under symplectic deformations.
In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian con…
The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.
problem Characterizing minimal and maximal surfaces in 3D and 3D-L spacetime.
method Analyzing surfaces with specific properties and using geometric and functional methods.
result Calabi-Bernstein type results for critical points of a weighted area functional in R3 and L3. For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, s…