arXiv research

Proves a conjecture for Calabi-Yau manifolds.

Simpler proof for quaternionic Calabi conjecture

Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.

Researchers solved a Calabi-Yau conjecture for specific 3D hypersurfaces in 4D space.

Solves a specific Calabi conjecture on special nilmanifolds.

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

Study explores unstable 3-forms on Calabi-Yau 3-folds.

Explores the geometric aspects of the Calabi Conjecture using PDE theory.

In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Ampère equation describes deformation of the Kähler (symplectic) structure.

This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we …

The paper proves a numerical condition for solving complex Hessian quotient equations with Calabi symmetry.

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces. In fact, we will prove considerably more. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 19…

Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.

Solves generalized Kähler Calabi-Yau problem on compact manifolds.

Sharp diameter bounds for Calabi-Yau degenerations proved.

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…

Proves SYZ conjecture for certain toric Fano hypersurfaces.

In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvatur…

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

Two applications of Aleksandrov-Bakelman-Pucci estimate to Calabi-Yau equation.

Estimates symplectic Calabi-Yau equation using Cheng-Yau method.

Survey on metric SYZ conjecture and non-archimedean geometry.

The abstract discusses special Lagrangians and their flow, proving conjectures and observing related phenomena.

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…

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…

Researchers prove birational invariance of BCOV invariant using motivic integration.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.

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.

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…

Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.

Paper proves solvability condition for complex equation on special submanifolds.

Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.

Extends arguments to limit structure in Calabi-Yau degenerations.

Mathematicians prove a conjecture about mirror symmetry at genus one.

Researchers create special fibrations on Calabi-Yau hypersurfaces.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $\int_{\partial P} u ~ d σ< C_1,$ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we sh…

Solves a conjecture about hyperKähler manifolds using quaternionic Monge-Ampère equation.

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…

Motivated by Strominger-Yau-Zaslow's mirror symmetry proposal and Kontsevich's homological mirror symmetry conjecture, we study mirror phenomena (in A-model) of certain results from Donaldson-Thomas theory for Calabi-Yau 4-folds.

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.

Sharp bounds for Calabi energy derived from geodesic rays and Mabuchi K-energy.

Extends BCOV invariant to pairs of Calabi-Yau manifolds and del Pezzo surfaces.

Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.

Study shows Ricci flat Calabi's metrics can't be projectively induced.