Mini-Workshop on Diﬀerential Geometry
and its Discretizations
Osaka City University (Building E of Faculty of Science, Lecture Room E408)
July 3–5, 2019Organizers
Joseph Cho (Kobe University)
Kosuke Naokawa (Hiroshima Institute of Technology)
Wayne Rossman (Kobe University)
Seong-Deog Yang (Korea University)
Masashi Yasumoto (chair, Osaka City University Advanced Mathematical Institute)Supports– JSPS Grant-in-Aid for Scientific Research on Innovative Areas, No.18H04489
Principal Investigator: Masashi Yasumoto, 2018-2020.
– Osaka City University Advanced Mathematical InsttituteInvited Speakers
Short Lectures (3 talks)
Wai Yeung Lam (Brown University, USA)
Katrin Leschke (University of Leicester, UK)

Invited Speakers
Joseph Cho (Kobe University, Japan)
Yoshiki Jikumaru (Kyushu University, Japan)
Masahiro Kawamata (Hiroshima University, Japan)
Callum Kemp (University of Bath, UK)Eunjoo Lee (Soongsil University, Korea)
Yuichiro Taketomi (Hiroshima University, Japan)

Program
July 3rd, 201913:50-14:00

Opening

14:00-14:50
Title
15:40-16:30
Title
17:00-17:50
Title

Wai Yeung Lam (Brown University)
Deformations of circle patterns
Yuichiro Taketomi (Hiroshima University)
On a Riemannian metric whose isometry group is maximal — a special caseKatrin Leschke (University of Leicester)
Isothermic surfaces via Quaternionic Holomorphic Geometry (1)July 4th, 201910:00-10:50
Title
11:20-12:10
Title
14:00-14:50
Title
15:40-16:30
Title
17:00-17:50
Title

Eunjoo Lee (Soongsil University)
Characterizations of the helicoid and its associate family
Masahiro Kawamata (Hiroshima University)
A construction of singular solutions for generalized Monge-Ampère equations
Yoshiki Jikumaru (Kyushu University)
Variational problem for anisotropic energy and its discretization
Wai Yeung Lam (Brown University)
Dimers and circle patterns
Katrin Leschke (University of Leicester)
Isothermic surfaces via Quaternionic Holomorphic Geometry (2)July 5th, 201910:00-10:50
Title
11:20-12:10
Title
14:00-14:50
Title
15:40-16:30
Title

Wai Yeung Lam (Brown University)
Weierstrass representation of discrete minimal surfaces
Joseph Cho (Kobe University)
Conserved quantities under Darboux transformations
Callum Kemp (University of Bath)Generalising the Darboux transform
Katrin Leschke (University of Leicester)
Isothermic surfaces via Quaternionic Holomorphic Geometry (3)16:30-16:35

ClosingTitle and Abstract
Wai Yeung Lam (Brown University)
Title 1: Deformations of circle patterns
Abstract
A circle pattern is a realization of a graph in the plane with cyclic faces, i.e. where all vertices
on a face lie on a circle. It is a central object in discrete conformal geometry. Following the ideas ofWilliam Thurston, two circle patterns with the same intersection angles are discretely conformallyequivalent.
We consider the deformation space of circle patterns and introduce discrete holomorphic quadratic
diﬀerentials. Their relation to the classical Teichmüller theory is discussed.
Title 2: Dimers and circle patterns
Abstract
The bipartite planar dimer model is the study of random perfect matchings (“dimer coverings”)
of a bipartite planar graph. We present a correspondence between the dimer model and circle
patterns, which holds for graphs that are either planar or embedded on the torus. In particular,
we focus on the cluster algebra structure and the Kasteleyn matrix. The latter is related to adiscretisation of the Dirac operator.
Title 3: Weierstrass representation of discrete minimal surfaces
Abstract
As an application of discrete holomorphic quadratic diﬀerentials, we present a discretisation ofminimal surfaces.

Katrin Leschke (University of Leicester)
Title: Isothermic surfaces via Quaternionic Holomorphic Geometry
Abstract
In this lecture series we will introduce QHG by using the example of isothermic surfaces.
Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. Wellknown examples include minimal surfaces, constant mean curvature surfaces but also surfaces ofrevolution. We will discuss classical transformations of isothermic surfaces, such as the T-transformor the Darboux transform, in the quaternionic calculus. Additionally, we generalise the associatedfamily of flat connections to allow for a complex parameter; in particular, we can generalise theclassical transforms. In case of a CMC surface, we have an additional family of flat connections
which is associated to the harmonic Gauss map of f . We discuss the link between these two
integrable systems.

Joseph Cho (Kobe University)
Title: Conserved quantities under Darboux transformations
Abstract
In this talk, we look at conserved quantities of flat connections that characterize the isothermicity of a surface. In particular, we review how they behave under Darboux transformations andrelate them to the discrete isothermic surface theory.

Yoshiki Jikumaru (Kyushu University)
Title: Variational problem for anisotropic energy and its discretization
Abstract
Anisotropic energy is the integral of an energy density that depends on the normal at eachpoint over the considered hypersurface in Rn+1 . The minimizer of such an energy among all closed
hypersurfaces enclosing the same (n + 1)-dimensional volume is unique and it is (up to rescaling)so-called the Wulﬀ shape. This energy is a generalization of the area and gives a mathematicalmodel of the energy for soap bubbles or crystals.
In this talk, we will explain a recent progress of our attempt which is a theory of discrete curves
and discrete surfaces based on the variational problem for anisotropic energy. As an application,
we visualize the anisotropic energy gradient flow by using the software JavaView.

Callum Kemp (University of Bath)Title: Generalising the Darboux transform
Abstract
The theory of isothermic surfaces in the conformal n-sphere may be viewed as part of a moregeneral theory of isothermic submanifolds in symmetric R-spaces. All the usual transforms havetheir analogues and we will see in particular how the analogue of the Darboux transform can becharacterised geometrically in terms of envelopes of certain submanifolds.

Masahiro Kawamata (Hiroshima University)
Title: A construction of singular solutions for generalized Monge-Ampère equations
Abstract
It is known that Monge-Ampère systems is a geometric formalization of Monge-Ampère equations using the theory of exterior diﬀerential systems. We introduce a generalization of MongeAmpère systems, Monge-Ampère equations and a relationship between such systems and equations.
Moreover we describe singular solutions for some generalized Monge-Ampère equations.

Eunjoo Lee (Soongsil University)
Title: Characterizations of the helicoid and its associate family
Abstract
The helicoid, along with the plane and the catenoid, is a classical example of minimal surfaces. Various characterizations have been known: a unique non-planar ruled minimal suruface, aunique complete, embedded, simply-connected non-planar minimal surface, etc. We characterize
a compact piece of the helicoid in a cylinder in R3 as an area-minimizing and a unique surface incertain geometrical restrictions. Furthermore, we investigate the associate family of helicoids andits uniqueness.

Yuichiro Taketomi (Hiroshima University)
Title: On a Riemannian metric whose isometry group is maximal — a special case
Abstract
In this talk, we study Riemannian metrics whose isometry groups are maximal with respect toinclusion. We get nice examples of such metrics by studying moduli spaces of left-invariant metrics
on some Lie groups.