Shih-Kai Chiu (邱詩凱)

I am now a Postdoctoral Scholar at Vanderbilt University working with Ioana Suvaina. Previously, I was a Postdoctoral Research Associate in Special Holonomy at the University of Oxford, where my mentor was Jason Lotay. I obtained my PhD in mathematics in August 2021 from the University of Notre Dame, under the supervision of Gábor Székelyhidi. I am interested in geometric analysis and complex differential geometry, as well as related areas.

Email: shih-kai.chiu@vanderbilt.edu

• Geometry and TACoS
• Oxford Geometry and Analysis Seminar ±
  Michaelmas 2021 - Trinity 2023. Co-organized with Guillem Cazassus, Ilyas Khan and Henry Liu.
• Student Seminar (Notre Dame) Fall 2020 ±
  Student geometry seminar held in Fall 2020 with a focus on talking about our own research.

Teaching

• Fall 2023, Math 2300 Multivariable Calculus, Instructor, Syllabus

Previous teaching ±

Research

I am currently interested in manifolds with special holonomy and calibrated submanifolds. My main focus is on Calabi-Yau manifolds and special Lagrangian submanifolds.

Papers & preprints

• Nonuniqueness of Calabi-Yau metrics with maximal volume growth, arXiv:2206.08210, submitted. ±
  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.
• (With G. Székelyhidi) Higher regularity for singular Kähler-Einstein metrics, arXiv:2202.11083, to appear in Duke Math. J. ±
  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.
• Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth, arXiv:1905.12965, to appear in Comm. Pure Appl. Math. ±
  We prove that on a complete Calabi-Yau manifold \(M\) with Euclidean 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 differential.
• On Calabi-Yau manifolds with maximal volume growth, PhD Thesis, Notre Dame, 2021. ±
  We study complete noncompact Calabi-Yau manifolds with maximal volume growth. These manifolds arise as desingularizations of affine Calabi-Yau varieties, and as bubbles in the singularity formation of compact Kähler-Einstein manifolds. The rather explicit metric behavior of these manifolds enables us to understand the metric behavior of the seemingly impenetrable compact case. It is therefore of great importance to study the uniqueness/deformation problem of these manifolds and metrics. Our first result is that subquadratic harmonic functions on such manifolds must be pluriharmonic. This should be seen as the linear version of the uniqueness/deformation problem. Next, we show that if a \(\partial\bar\partial\)-exact Calabi-Yau manifold with maximal volume growth has tangent cone at infinity which splits an Euclidean factor, then the metric is unique under subquadratic perturbation of the Kähler potential. Geometrically, this means that the Calabi-Yau metric is rigid under deformations that fix the tangent cone at infinity. It is therefore interesting to know if a quadratic perturbation of the Kähler potential, which should correspond to an automorphism of the tangent cone at infinity, really changes the metric. As a first step toward understanding this picture, we construct new Calabi-Yau metrics on \(\mathbf{C}^3\) with tangent cone at infinity given by \(\mathbf{C} \times A_2\), and we show that these metrics are inequivalent in the sense that they are not related by isometries and scalings. This existence result provides the first example of families of Calabi-Yau metrics with the same tangent cone at infinity while the underlying complex structure is fixed.

Invited Talks

Conferences

• Several Complex Variables, Complex Geometry and related PDEs, Maryland, 05/2024
• Analytic Methods in Complex Geometry, Münster, 08/2023
• Connections between String Theory and Special Holonomy, Oxford, 01/2022

Seminars

• Geometry, Topology, Dynamical Systems Seminar, UT Dallas, 10/2023
• Geometry and Geometric Analysis Seminar, Purdue, 10/2023
• Geometry Seminar, Vanderbilt, 09/2023
• Geometry and Topology Seminar, Glasgow, 05/2023
• Geometry and Topology Seminar, UQAM, 01/2023
• Geometric Analysis Seminar, ITS Westlake, 11/2022
• Geometric Analysis Seminar, BICMR, 10/2022
• Geometric Analysis Seminar, CUHK, 10/2022
• Informal Complex Geometry and PDE Seminar, Columbia, 10/2022
• Geometry Seminar, King's College London, 10/2022
• Geometry and Analysis Seminar, Oxford, 11/2021
• Differential Geometry Seminar, NCTS, 08/2021
• Geometric Analysis Seminar, Notre Dame, 11/2020
• Oberseminar Differentialgeometrie, Münster, 07/2020
• Geometric Analysis Seminar, CUNY, 06/2020
• Differential Geometry Seminar, NCTS, 06/2019
• Geometric Analysis Seminar, Notre Dame, 04/2019