Related papers: Higher-order estimates for collapsing Calabi-Yau m…

We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kahler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end we prove an asymptotic…

As a sequel to \cite{Licollapsing}, we study Calabi-Yau metrics collapsing along a holomorphic fibration over a Riemann surface. Assuming at worst canonical singular fibres, we prove a uniform diameter bound for all fibres in the suitable…

We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space…

We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.

We establish a $C^{1,\alpha}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,\alpha}$ harmonic radius, where we use the blow up analysis as in…

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these…

Let $(\mathscr{C}, \omega_{\mathscr{C}})$ be a Ricci-flat, simply connected, conical K\"ahler manifold. We establish a Liouville theorem for constant scalar curvature K\"ahler (cscK) metrics on $\mathscr{C}$. The theorem asserts that any…

Motivated by the study of collapsing Calabi-Yau threefolds with a Lefschetz K3 fibration, we construct a complete Calabi-Yau metric on $\mathbb{C}^3$ with maximal volume growth, which in the appropriate scale is expected to model the…

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the…

We study polarised algebraic degenerations of Calabi-Yau manifolds. We prove a uniform Skoda type estimate, and a uniform $L^\infty$-estimate for the Calabi-Yau K\"ahler potentials.

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration…

We define the counting of holomorphic cylinders in log Calabi-Yau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new…

Let $X$ denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let $D$ be any fiber of the resulting elliptic fibration on $X$. Using ansatz metrics inspired by work of Gross-Wilson and a PDE method…

We present new complete Calabi-Yau metrics defined on the complement of a smooth anticanonical divisor with ample normal bundle, approaching the Calabi model space at a polynomial rate. Moreover, we establish the uniqueness of this type of…

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a…

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a…

We study certain polarized degenerations of Calabi-Yau manifolds near an intermediate complex structure limit, and improve the potential $C^0$-convergence to a metric convergence result on the generic region for the corresponding collapsing…

We give sharp $C^{2,\alpha}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous…

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ in dimensions $n\ge 3$. These conjectures ask whether a complete minimal hypersurface must be unbounded, and more…