RIMS Workshop (Type A)

Applications of Harmonic Maps and Higgs Bundles
to Diﬀerential Geometry
RIMS Kyoto University and Online (Zoom)
May 28 – June 2, 2022
Supports– Research Institute for Mathematical Sciences(Kyoto University, International Joint Usage/Research Center)– JSPS Grant-in-Aid for Scientiﬁc Research (A) No.19H00636
Principal Investigator: Kaoru Ono– JSPS Grant-in-Aid for Scientiﬁc Research (C) No.18K03307
Principal Investigator: Yoshihiro Ohnita– JSPS Grant-in-Aid for Early-Career Scientists No.20K14314
Principal Investigator: Masashi Yasumoto

OrganizersToru Kajigaya (Tokyo University of Science)
Katrin Leschke (University of Leicester & RIMS)
Yoshihiro Ohnita (Deputy Chair, OCAMI Osaka Metropolitan University)
Kaoru Ono (RIMS Kyoto University)
Franz Pedit (Chair, UMass Amherst & RIMS)
Wayne Rossman (Kobe University)
Masashi Yasumoto (Deputy Chair, Tokushima University)

Contact
Toru Kajigaya (kajigaya@rs.tus.ac.jp)
Yoshihiro Ohnita (ohnita@osaka-cu.ac.jp)
Masashi Yasumoto (yasumoto.masashi@tokushima-u.ac.jp)

RIMS Workshop (Type A)

Applications of Harmonic Maps and Higgs Bundles
to Diﬀerential Geometry
RIMS Kyoto University and Online (Zoom)
May 28 – June 2, 2022
Program
May 28, 2022
14:00-15:15
15:50-16:50

John Loftin (Rutgers University, USA)Diﬀerential geometry of cubic diﬀerentials (1)Shoichi Fujimori (Hiroshima University, Japan)
Genus three embedded doubly periodic minimal surfaces with parallelends

May 29, 2022
9:30-10:45
11:10-12:10
14:00-15:15
15:50-16:50

Martin Traizet (University of Tour, France)
Gluing constructions via loop group methods
Hiroki Fujino (Nagoya University, Japan)
Extension of harmonic function beyond discontinuous boundary values
Fran Burstall (University of Bath, UK)Harmonic maps, Gauss maps and classical geometry (1)Natsuo Miyatake (Osaka University, Japan)
Kazdan-Warner type equations on Riemannian manifolds, theHermitian-Einstein equation for diagonal metrics on Higgs bundles, andthe variational method

May 30, 2022
9:30-10:45
11:10-12:10
12:10-

John Loftin (Rutgers University, USA)Diﬀerential geometry of cubic diﬀerentials (2)Yu Kawakami (Kanazawa University, Japan)
Heinz-type mean curvature estimates and its application
Free Discussion

May 31, 2022
9:30-10:45
11:20-12:00

14:00-15:15

15:50-16:50

Fran Burstall (University of Bath, UK)
Harmonic maps, Gauss maps and classical geometry (2)Tadashi Udagawa (Waseda University, Japan)
Globality of the DPW construction for Smyth potentials in the caseof SU1,1
Martin Traizet (University of Tour, France)
Loop group methods for the non-abelian Hodge correspondence on
a 4-punctured sphere
Martin Guest (Waseda University, Japan)
Toda and Higgs

June 1, 2022
9:30-10:45

11:10-12:10

14:00-15:15
15:15-15:25

Franz Pedit (UMass Amherst, USA & RIMS Kyoto University, Japan)
Higgs bundles, aﬃne spheres, Monge-Ampère equations, and SYZ
mirror symmetry
Isami Koga (Meiji University, Japan)
Equivariant harmonic immersions of the complex projective line into
the complex Grassmannians of two-planes
Charles Ouyang (UMass Amherst, USA)
Maximal surfaces in H2,2 and their limits
Closing Address

June 2, 2022
10:00-16:00

Free Discussion

RIMS Workshop (Type A)

Applications of Harmonic Maps and Higgs Bundles
to Diﬀerential Geometry
RIMS Kyoto University and Online (Zoom)
May 28 – June 2, 2022
Abstracts
Fran Burstall (University of Bath, UK)
Title: Harmonic maps, Gauss maps and classical geometry
Abstract
Harmonic maps ﬁnd application as the Gauss maps (in an appropriate sense) of solutions ofclassical variational problems. Constant curvature surfaces in R3 ; Willmore surfaces, projectively
minimal and Lie sphere minimal surfaces all ﬁt into this picture giving a uniform approach to theirintegrability and Baecklund transformations. In these talks, I shall survey this topic and indicate aunifying approach to the latter three surface classes via the classical notion of the Laplace invariantof a line congruence.

John Loftin (Rutgers University, USA)
Title: Diﬀerential geometry of cubic diﬀerentials
Abstract
Cubic diﬀerentials are related to diﬀerential geometric structures on surfaces with symmetricgroup a real form of SL(3, C). In my ﬁrst lecture, I will discuss some of the geometry and analysis
behind four natural cases. We will focus on 2 cases, hyperbolic aﬃne spheres in 3-space andminimal Lagrangian surfaces in the complex hyperbolic plane and explain their relation to Higgs
bundles, harmonic maps, and surface group representations.
In the second lecture, I plan to discuss an important case of limits of hyperbolic aﬃne spheres.
Pairs (S, U ) of a Riemann surface S of genus g and U a cubic diﬀerential on S can be used toparametrize the Hitchin component of the space of representations of the fundamental group into
PSL(3, R). For nonzero U , we study the case sU as s approaches inﬁnity. In particular, we show
the geometry in this limit can be read oﬀ explicitly from U , in terms of an embedding of theuniversal cover of S into the real building given by the asymptotic cone of the symmetric spaceSL(3, R)/SO(3). This is based on joint work with Andrea Tamburelli and Mike Wolf.

Martin Traizet (University of Tours, France)
Title (1): Gluing constructions via loop group methods
Abstract
Loop group methods (e.g. the DPW method) allow the construction of constant mean curvaturesurfaces from holomorphic data. For surfaces with non-trivial topology, a monodromy problem
must be solved. I will explain some ideas on how to solve the monodromy problem by an implicit
function argument at a point (in the parameter space) where the surface degenerates. With theseideas, we can carry on gluing constructions classically done using PDE methods.
Title (2): Loop group methods for the non-abelian Hodge correspondence on a 4-punctured
sphereAbstract
The non-abelian Hodge correspondence is a real analytic map between the moduli space ofHiggs bundles and the De Rham moduli space of ﬂat connections, mediated by solutions of theself-duality equation. Using some of the ideas in the ﬁrst talk, I will explain how to constructsolutions of the self-duality equation for Higgs ﬁelds on a 4-punctured sphere with small parabolicweights. This allow us to compute quite explicitly the non-abelian Hodge correspondence in thatparticular case. Joint work with L. Heller and S. Heller.

Shoichi Fujimori (Hiroshima University, Japan)
Title: Genus three embedded doubly periodic minimal surfaces with parallel ends
Abstract
We construct a one-parameter family of embedded doubly periodic minimal surfaces in Euclidean three space. Each surface of the family has genus three with four parallel ends in thequotient. This is joint work with Peter Connor (Indiana University South Bend) and PhillipMarmorino (University of Notre Dame).

Hiroki Fujino (Nagoya University, Japan)
Title: Extension of harmonic function beyond discontinuous boundary values
Abstract
The classical Schwarz reﬂection principle induces some refection principles for minimal surfaces in the Euclidean space. Such reﬂection principles enable us to construct various types ofcomplicated and beautiful periodic minimal surfaces in an intuitive manner.

In this talk, we introduce a method of real analytic extension for harmonic functions beyond
their discontinuous boundary values. In addition, as an application of the result, we constructsome kinds of periodic zero mean curvature surfaces. This talk is based on the joint research withShintaro Akamine (Nihon University).

Martin Guest (Waseda University, Japan)
Title: Toda and Higgs
Abstract
The 2D Toda equations can take several forms, most of which can be related to harmonic
map equations for maps from a surface to a symmetric space. It turns out that one of theseforms can be related to the Higgs equations (Hitchin’s equations), and in this situation the DPW
potential plays the role of the Higgs ﬁeld. We have studied in some detail the case where the Higgs
ﬁeld is given by monomials; this is analogous to case of the the Smyth potential in the versionof the Toda equations corresponding to CMC surfaces. Takuro Mochizuki has shown that thenonabelian Hodge Correspondence applies in this situation, even when singularities are allowed in
the potential (in contrast to the CMC situation, where there are no singularities). We shall discuss
some diﬀerential geometric properties of the resulting harmonic maps.

Yu Kawakami (Kanazawa University, Japan)
Title: Heinz-type mean curvature estimates and its application
Abstract
We would like to talk about a systematic study of Heinz-type mean curvature estimates in
Euclidean space and Lorentz-Minkowski space. In particular, we give a uniﬁed description ofHeinz-type mean curvature estimates under an assumption on the gradient bound for space-likegraphs and time-like graphs in Lorentz-Minkowski space and its application to Bernstein-type
theorem of entire graphs of constant mean curvature. This talk is based on a joint work withAtsufumi Honda, Miyuki Koiso and Syunsuke Tori.

Isami Koga (Meiji University, Japan)
Title: Equivariant harmonic immersions of the complex projective line into the complex
Grassmannians of two-planes
Abstract
In this talk, I would like to introduce a classiﬁcation result of equivariant harmonic immersios
of complex projective line into the complex Grassmannian of two-planes. To classify them, at ﬁrst
we research some properties of invariant connections on homogeneous vector bundle of rank twoover the complex projective line. Then we give the main result.

Natsuo Miyatake (Osaka University, Japan)
Title: Kazdan-Warner type equations on Riemannian manifolds, the Hermitian-Einstein equation
for diagonal metrics on Higgs bundles, and the variational methodAbstract
We can consider the Hermitian-Einstein equation for diagonal metrics on Higgs bundles overa Kähler manifold as one of the Kazdan-Warner type PDE in addition to the view that it is theHermitian-Einstein equation. Here we mean by a diagonal metric a Hermitian metric on a direct
sum of holomorphic line bundles diagonal to the decomposition and by Kazdan-Warner equation
the Poisson’s equation with an exponential non-linear term. In this talk, we formally generalize
the Hermitian-Einstein equation for diagonal metrics on Higgs bundles to PDEs on Riemannian
manifolds and solve them when the manifold is compact using the variational method inspired by
the Kempf-Ness’s theorem for linear torus actions.

Charles Ouyang (UMass Amherst, USA)
Title: Maximal surfaces in H2,2 and their limits
Abstract
Equivariant conformal harmonic maps from the universal cover of a closed Riemann surface
into a non-compact symmetric space are useful in ascribing geometric meaning to surface grouprepresentations into Lie groups via the non-Abelian Hodge correspondence. After brieﬂy reviewing instances of this for the groups G = PSL2 R × PSL2 R and SL3 R, we discuss G = Sp4 R, where
by work of Collier-Tholozan-Toulisse, the geometric objects are given by equivariant space-likemaximal surfaces in the pseudo-hyperbolic space H2,2 . We shall describe the limiting behavior ofdegenerate maximal surfaces, and how these interact when the representations are taken from different connected components of the Sp4 R-character variety. This is joint work with A. Tamburelli.

Franz Pedit (UMass Amherst, USA)
Title: Higgs bundles, aﬃne spheres, Monge-Ampère equations, and SYZ mirror symmetry
Abstract
We construct special Lagrangian 3-torus ﬁbrations, singular over a trivalent discriminant locus,of Calabi-Yau 3-folds using parabolic non-Abelian Hodge theory. This is an elaboration of an
approach ﬁrst suggested in a paper by Loftin, Yau, and Zaslow.

Tadashi Udagawa (Waseda University, Japan)
Title: Globality of the DPW construction for Smyth potentials in the case of SU1,1
Abstract
In this talk, we construct harmonic maps from Smyth potentials()
0 z k01
ξ=dz, λ ∈ S 1 , k0 , k1 ∈ Z,k1
λ z0on C∗ by using the DPW method. The DPW method is a way to construct a CMC-surface, thatwas developed by J. Dorfmeister, F. Pedit and H. Wu for studying harmonic maps into symmetricspaces. The Gauss map of a constant mean curvature (CMC) surface in R3 or R2,1 is a harmonic
map, thus the DPW method is useful for studying constant mean curvature (CMC)-surfaces. InR2,1 case, the Gauss-Codazzi equation can be regarded as the sinh-Gordon equation
utt = e2u − e−2u , u : U → R, U ⊂ C.
In the DPW method, u is obtained from the Iwasawa factorization of a solution L of L−1 dL = ξin the case of SU1,1 . However, the Iwasawa factorization in noncompact case is not always global.
Thus, the globality of the Iwasawa factorization is an important problem. In [1], J. Dorfmeister,M. Guest and W. Rossman gave the DPW construction for Smyth potential near z = 0. In thistalk, we show that L can be expressed in terms of Bessel functions and from the asymptotic
expansion of Bessel functions at large z we solve a Riemann-Hilbert problem to give a global
Iwasawa factorization. In this talk, for simplicity, we consider the case k0 = 3, k1 = −1.

References[1] J. F. Dorfmeister, M. A. Guest, and W. Rossman, The tt∗ structure of the quantum cohomologyof CP 1 from the viewpoint of diﬀerential geometry, Asian J. Math. 14 (2010) 417–438.