Shih-Kai Chiu
Since Winter 2025, I am a Visiting Assistant Professor
at University of
California, Irvine, where my mentor
is Xiangwen
Zhang. During Fall 2024, I was a Postdoc SLMath at
the Simons Laufer
Mathematical Sciences Institute (formerly MSRI) under
the
program Special
Geometric Structures and Analysis, where my mentor was
Mark
Haskins.
During the academic year 2023-2024, I was a Postdoctoral
Scholar
at Vanderbilt
University working
with Ioana
Suvaina. During the academic years 2021-2023, I was a
Postdoctoral Research Associate in Special Holonomy at
the University of
Oxford, where my mentor
was Jason D.
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:
shihkaic@uci.edu
Teaching
- Spring 2025, Math 299C Reading Course (Dirac Operators and Index Theory)
- Spring 2025, Math 3A Introduction to Linear Algebra, Instructor
- Spring 2025, Math 2E Multivariable Calculus II, Instructor
- Winter 2025, Math 2E Multivariable Calculus II, Instructor
- Winter 2025, Math 240B Differential Geometry, Instructor
Past
teaching ±
At Vanderbilt:
- Spring 2024, Math 2400 Differential Equations with Linear Algebra, Instructor
- Fall 2023, Math 2300 Multivariable Calculus, Instructor, Syllabus
At Notre Dame:
- Spring 2021, MATH 10560 Calculus II, Teaching Assistant
- Fall 2019, MATH 10550 Calculus I, Instructor
- Fall 2018, MATH 10350 Calculus A, Teaching Assistant
- Spring 2018, MATH 20580 Introduction to Linear Algebra and Differential
Equations, Teaching Assistant
- Fall 2017, MATH 10550 Calculus I, Teaching Assistant
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
- (With Y. Chen, M. Hallgren, G. Székelyhidi, T. D. Tô,
F. Tong) On Kähler-Einstein
currents, arXiv:2502.09825.
±
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.
- (With Y.-S. Lin) Special Lagrangian submanifolds in
K3-fibered Calabi-Yau
3-folds, arXiv:2410.17662.
±
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.
- Nonuniqueness of Calabi-Yau metrics with maximal volume growth,
arXiv:2206.08210.
±
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,
Duke Math. J. 172(18): 3521-3558 (1 December 2023).
±
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,
Open Access,
Comm. Pure Appl. Math. 77 (2024), no. 6, 3080-3106. ±
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 and workshops
- New directions in G2 geometry, AIM Workshop, 03/2025
- PDE methods in Complex Geometry, AIM Workshop, 08/2024
- Geometry Crossing Singularities, TIMS, Taipei, 07/2024
- Several Complex Variables, Complex Geometry and related PDEs, Maryland, 05/2024
- 2023 CMS Winter Meeting, Montreal, 12/2023
- Analytic Methods in Complex Geometry, Münster, 08/2023
- Connections between String Theory and Special Holonomy, Oxford, 01/2022
Seminars
- Differential Geometry Seminar, UC Irvine, 05/2025
- Differential Geometry Seminar, UC Santa Barbara, 04/2025
- Geometry and Topology Seminar, UC Riverside, 04/2025
- Informal Geometric Analysis Seminar, Northwestern, 02/2025
- GSA Seminar, SLMath, 09/2024
- Informal Complex Geometry Seminar, Columbia, 05/2024
- Geometric Analysis Seminar, CUHK, 12/2023
- 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, University of 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