Shih-Kai Chiu
My name is Shih-Kai Chiu (邱詩凱). I am a Postdoctoral
Research Associate in Special Holonomy at
the University of
Oxford, where my mentor
is 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.
Write me
an email.
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
- Analytic Methods in Complex Geometry, Münster, 08/2023
- Connections between String Theory and Special Holonomy, Oxford, 01/2022
Seminars
- Geometry and Topology Seminar, UQAM, 1/2023
- 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
Teaching
The following courses were taught at Notre Dame.
- MATH 10560, Spring 2021, Teaching Assistant
- MATH 10550, Fall 2019, Instructor
- MATH 10350, Fall 2018, Teaching Assistant
- MATH 20580, Spring 2018, Teaching Assistant
- MATH 10550, Fall 2017, Teaching Assistant