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Shih-Kai Chiu

Publications and source records attributed to Shih-Kai Chiu.

8 recordsLinked to original sources
arXiv2026

Nowhere-vanishing harmonic 1-forms on real loci of K3-fibred Calabi-Yau 3-folds

We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat Kähler metrics. We apply our construction to examples whose real loci have connected components diffeomorphic to $S^1\times S^2$ and to both trivial and nontrivial mapping tori. As an application, we produce examples of compact $7$-manifold with holonomy $G_2$ via the Joyce-Karigiannis construction.

math.DG↗
arXiv2025

From tropical curves to special Lagrangians

We show that any locally planar tropical curve $Γ\subset \mathbb{R}^n$ (with unit edge weights) can be realized as the limit of the rescaled moment map images of a family of special Lagrangian submanifolds in $T^*T^n$ with respect to the Euclidean structure. This is based on a gluing construction that matches special Lagrangian local models to the combinatorics of $Γ$, thereby establishing a direct link between tropical geometry and special Lagrangian geometry.

math.DG↗
arXiv2025

On Kähler-Einstein Currents

We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in $L^p$ for $p > \frac{2n-1}{n}$, then the metric defines an RCD space.

math.DG↗
arXiv2024

Special Lagrangian submanifolds in K3-fibered Calabi-Yau 3-folds

We construct special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces. As these 3-folds collapse, the special Lagrangians shrink to 1-dimensional graphs in the base, mirroring the conjectured tropicalization of holomorphic curves in collapsing SYZ torus-fibered Calabi-Yau manifolds. This confirms predictions of Donaldson and Donaldson-Scaduto in the Calabi-Yau setting. Additionally, we discuss our results in the contexts of the Thomas-Yau conjecture, the Donaldson-Scaduto conjecture, and mirror symmetry.

math.DG↗
arXiv2022

Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth

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 $L^2$ estimate of the exterior derivative.

math.DG↗
arXiv2022

Nonuniqueness of Calabi-Yau metrics with maximal volume growth

We construct a family of inequivalent Calabi-Yau metrics on $\mathbf{C}^3$ asymptotic to $\mathbf{C} \times A_2$ at infinity, in the sense that any two of these metrics cannot be related by a scaling and a biholomorphism. This provides the first example of families of Calabi-Yau metrics asymptotic to a fixed tangent cone at infinity, while keeping the underlying complex structure fixed. We propose a refinement of a conjecture of Székelyhidi addressing the classification of such metrics.

math.DG↗
arXiv2022

Higher regularity for singular Kähler-Einstein metrics

We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete $\partial\bar\partial$-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.

math.DG↗