Geometry of Moduli Space of Low Dimensional
Manifolds
RIMS, Kyoto University
December 14–18, 2016

Supported in part by JSPS Grant-in-aid for Scientific Research
No.24340009 基盤研究 (B)
低次元多様体モジュライ空間の新展開（代表者：山田澄生）

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Monday, December 12

Free boundary minimal surfaces I & II
Martin Li
Chinese University of Hong Kong
Abstract: The purpose of these two talks is to give a survey of free boundary
minimal surfaces in Riemannian three-manifolds with boundary. Part I of the talkwill cover elementary notions of free boundary minimal surfaces including theirfirst and second variations. We will describe some classical existence and regularityresults. In Part II, we will describe some more recent advances in the field, withparticular emphasis on the Euclidean unit ball and also a general existence theory
by min-max constructions.

Interrogating length spectra and quantifying isospectral
finiteness I & II
Hugo Parlier
University of Fribourg
Abstract: Associated to a closed hyperbolic surface is its length spectrum, the setof the lengths of all of its closed geodesics. Two surfaces are said to be isospectral
if they share the same length spectrum. There are diﬀerent methods to produce
surfaces that are isospectral but not isometric, the most successful one based on atechnique introduced by Sunada.
The talks will be about the following questions and how they relate.
• How many questions do you need to ask to determine a length spectrum?
• In a given genus how many diﬀerent surfaces can be isospectral but not isometric?
The approach to these questions will include finding adapted coordinate sets formoduli spaces and exploring McShane type identities.

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Tuesday, December 13
Kontsevich’s eye and associators I & IIHidekazu Furusho
Nagoya University
Abstract: In my talk, I will talk about a new associator constructed by Alekseev
and Torossian by reviewing a way to compactify certain “configuration spaces" andthe definition of Kontsevich’s weights of Lie graphs.

Three-holed sphere groups in PGL(3, R), I
Sungwoon KimJeju National University
Abstract: The notion of Anosov representation is a generalization of convex cocompact representation in hyperbolic 3-space to higher rank symmetric spaces. In thistalk we shall introduce this notion by looking at typical examples in the PGL(3, R)
character variety.
For example, hyperconvex representations introduced by Labourie and horocyclic representations introduced by Barbot when the domain group is the surface
group, and positive representations introduced by Fock and Goncharov, and Pappus
representations introduced by Schwartz when the domain group is the free group(of rank 2).
While discussing these examples, we shall focus on the property of boundaryembeddedness (slightly weaker notion than Anosov) with our motivational questions
in mind for the second talk.
(Work in progress joint with Jaejeong Lee)

Three-holed sphere groups in PGL(3, R), II
Jaejeong Lee
Korea Institute of Adanace Study
Abstract: As shown in the first talk, among Anosov representations in the PGL(3, R)
character variety of the free group, there are representations that are negative ina certain sense, as well as the well-known positive (hyper-convex) representations.
In this second talk, we focus on such representations which come from the framed
representations (in the sense of Fock-Goncharov) of the 3-holed sphere group.
In order to exhibit concrete examples we restrict our attention to the (real 2dimensional) relative character variety where the boundary monordomies are allquasi-unipotent. Boundary-embedded and transversal (or antipodal) representations in this subvariety may be called relatively Anosov and all such examples properly include the Pappus representations studied by R. Schwartz in 1993.
We further restrict to a certain real 1-dimensional subvariety consisting of representations with 2-fold symmetry and find all boundary-embedded ones in there,
thereby obtaining a PGL(3, R) analogue of R. Schwartz’s solution to the GoldmanParker conjecture on the ideal triangle reflection groups in PU(2, 1).
(Work in progress joint with Sungwoon Kim)
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Wednesday, December 14

Geometric equivalence among smooth section-germs
of vector bundles with respect to structure groups I & II
Shyuichi Izumiya
Hokkaido University
Abstract: We consider equivalence relations among smooth section-germs of vector bundles with respect to structure groups. One of these equivalence relations
is a slight generalization of G-equivalence among smooth map germs introducedby Tougeron [6]. Altough Gervais [1, 2, 3] investigated some properties of Gequivalence, there are no proper examples in their papers. We give several interesting applications including quantum chemistry and spintronics [5], which areexpected to have an application to the theory of topological insulators and so on.
Our equivalence relation is a slight generalization of G-equivalence. However, weemphasize that we can open a new door for some important applications of singularity theory. Moreover we define other equivalence relations with respect to structuregroups of vector bundles.
Then we discover that there are a lot of previous applications or new applications of the theory of singularities as examples of these equivalence relations. In
this talk I will start to explain the celebrating results of Mahter [4] and its generalizations by Tougeron for non-speciaists.The goal of my talk is an explanation of
the moduli space to describe the diﬀerence of Mahter’s K or A-equivalence between
our geometric equivalence.

References[1] J. J. Gervais, Suﬀuciency of jets. Pacific Journal of Mathematics, 72, 419–422,(1977)[2] J. J. Gervais, G-stability of mappings and stability of bifurcation diagrams. J.
London Math. Soc. 25, 551–563, (1982)[3] J. J. Gervais, Deformations G-verselles et G-stables. Can. J. Math. XXXVI,
9–21, (1984)[4] J. Mather, Stability of C ∞ -Mappings III. Finitely Determined Map Germs. Publ.
Math. I.H.E.S. 36 , 127–156, (1968)[5] H. Teramoto, K. Kondo, S. Izumiya, M. Toda and T. Komatsuzaki, Classificationof Hamiltonians in neighborhoods of band crossings in terms of the theory ofsingularities. Preprint (2016)[6] J. C. Tougeron, Ideaux fonctions diﬀerentiables. Springer-Verlag, Berlin, Heiderberg, New York 1972

4Thursday, December 15

On the geometry and topology of initial data sets inGeneral Relativity I & IIGreg Galloway
University of Miami

Abstract: An initial data set in spacetime consists of a spacelike hypersurface V,
together with its induced (Riemannian) metric h and its second fundamental form
K. After a brief introduction to spacetime geometry and general relativity, we will
present some topics of recent interest related to the geometry and topology of initial
data sets. In particular, after reviewing Hawking’s theorem on black hole topology,
we will consider the topology of black holes in higher dimensional gravity, inspiredby certain developments in string theory and issues related to black hole uniqueness.
We shall also discuss recent work on the geometry and topology of the regionof space exterior to all black holes, which is closely connected to the notion oftopological censorship. Many of the results to be discussed rely on the recently
developed theory of marginally outer trapped surfaces, which are natural spacetime
analogues of minimal surfaces in Riemannian geometry.
These talks are partially based on work with a number of collaborators: Lars
Andersson, Mattias Dahl, Michael Eichmair, Dan Pollack and Rick Schoen.

5Inverse scattering on non-compact manifolds
with general metric I & II
Hiroshi Isozaki
University of Tsukuba

Abstract: The problem we address is the inverse scattering on non-compact Riemannian manifolds (or orbifolds, or more generally, manifolds with conical singularities) M consisting of a union of open sets :M = K ∪ M1 ∪ · · · ∪ MN +N ′ ,where K is compact, Mi is diﬀeomorphic to (1, ∞) × Mi , Mi being a compact n − 1
dimensional manifold (or orbifold) endowed with the metric hMi . On each end Mi ,the metric of M is assumed to behave likeds2 ∼ (dr)2 + ρi (r)2 hMi ,r → ∞.

Typical examples of ρi (r) are(ρi (r) = A exp c0 r +)βr−αwith 0 < α < 1,1−α( )or A exp c0 r rβ .

(1)

We assume that for 1 ≤ i ≤ N , the end Mi has a regular infinitiy, i.e. in (1) c0 ≥ 0and if c0 = 0, then β > 0, while for N + 1 ≤ i ≤ N + N ′ , it has a cusp, i.e. c0 ≤ 0and if c0 = 0, then β < 0. So, it includes ends such as regular infinities of hyperbolic
manifolds : ρ(r) = er , euclidean ends : ρ(r) = r, cusps of hyperbolic manifolds :ρ(r) = e−r .
By observing the asymptotic behavior at infinity of generalized eigenfunctions for
the Laplace operator on M, we introduce the S-matrix (phyiscal S-matrix for regularends and generalized S-matrix for cusps), and then solve the inverse scattering
problem, i.e. the recovery of the manifold M from one component of the S-matrix(for all energies).
This is a joint work with Y. Kurylev and M. Lassas.

6Friday, December 16

Neck-Pinching of CP 1 -structures.
Shinpei Baba
University of Heidelberg
Abstract: We discuss about a certain geometric structure (locally homogeneousstructure) on a surface and its degeneration. A CP 1 -structure on a surface is anatlas modeled on CP 1 with transition maps in P SL(2, C), and it can be regarded
as a Riemann surface with a holomorphic quadratic diﬀerential. Moreover each
CP 1 -structure has a holonomy representation from the fundamental group of thesurface into P SL(2, C).
Given a diverging one-parameter family of CP 1 -structures such that its holonomy representation converges, we would like to characterize it divergence. We
discuss about its limit in the special case when its conformal structure is “pinched”
along a loop.

Rational represerentation of mapping class group acting on
Teichmüller space I & II
Toshihiro Nakanishi
Shimane University
Abstract:
We show that a tuple of 6g−5 geodesic length functions defined on the Teichmüller
space of genus g ≥ 2 gives a coordinate-system of the Teichmüller space and theaction of the mapping class group on it can be represented by a group of rationaltransformations in this coordinate-system. The rational representation has severalapplications. One of them is a list of presentations by Dehn twists of all finite
subgroups of the mapping class group of genus 2 up to conjugacy.
This is a jint work with Gou Nakamura.

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