| Title and Abstract of Talks：
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| Andreas Arvanitoyeorgos |
| Title: Homogeneous Einstein manifolds. An Overview and Recent Results |
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| Abstract:
A Riemannian manifold $(M, g)$ is called Einstein if $\Ric(g) = c g$ for some $c\in \bb{R}$.
For a homogeneous space $G/H$ the problem is to prove existence of a $G$-invariant metric and
if possible find all invariant Einstein metrics (up to scale and isometry).
I will restrict to the case when $c>0$ ($G/H$ is compact) and give an overview of recent results
for two major classes of homogeneous spaces.
For those whose isotropy representation $\chi$ decomposes into a direct sum of
irreducible and {\it non equivalent} subrepresentations, and those for which
$\chi$ contains {\it equivalent} subrepresentations.
In the last case the description of $G$-invariant metrics is more complicated,
which makes the problem of proving existence of invariant Einstein metrics more complicated.
Typical examples in the first class of homogeneous spaces are the generalized flag manifolds,
and in the second class the Stiefel manifolds (real, complex, or quaternionic).
I will also discuss results about Einstein metrics on homogeneous spaces examples of which
belong to both classes, such as generalized Wallach spaces.
The case of finding left-invariant Einstein on compact Lie groups requires
a special attention and we refer to M. Statha's talk about this.
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| Anna Gori |
| Title: The moment map: a powerful tool in understanding submanifolds geometry |
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| Abstract:
Whenever a compact Lie group G acts on a symplectic manifold M in a Hamiltonian fashion
it is possible to define a moment map from the manifold to the dual of the Lie algebra of G.
The aim of the talk is to present several results attained during the past few years by dealing extensively
on the properties of such an important function.
Namely the following three problems will be addressed:
(a) existence of Homogeneous Lagrangian submanifolds in compact Kaehler manifolds with h^[1,1}=1,
(b) minimality of Lagrangian submanifolds in Kaehler Einstein manifolds;
(c) isometrical embeddings in complex projective spaces of submanifolds admitting Kaehler Ricci Solitons .
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| Giuseppe Pipoli |
| Title:
Mean curvature flow of pinched submanifolds in the complex projective space and in the sphere.
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| Abstract:
The study of the mean curvature flow of pinched submanifold starts in 1987 with Huisken's paper
about the evolution of hypersurfaces of the sphere.
I will describe an extension of this result to pinched submanifolds of complex projective space and,
using the commutativity of the mean curvature flow with the Hopf fibration, to new examples in the sphere.
As a consequence we will prove a classefication result for the submanifolds considered.
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| Young Jin Suh
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| Title: Real hypersurfaces with isometric Reeb flow in Hermitian Symmetric Spaces |
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| Abstract: We investigate real hypersurfaces with isometric Reeb flow in Hermitian symmetric spaces.
In particular, we give some classifications of real hypersurfaces with isometric Reeb flow
in generalized compact complex k-palne Grassmannians and non-compact complex hyperbolic quadric.
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| Byung Hak Kim |
| Title:On Conformal transformation related Ricci curvature conditions |
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In this talk, we are to report recent resullts of the Riemannian manifolds admitting conformal transformations
with various Ricci curvature conditions. Moreover we introduce and consider about the open problems related this topic.
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| Makiko Sumi Tanaka |
| Title: Maximal antipodal sets of the bottom space of Sp(n)/U(n) |
| Abstract:
We classified maximal antipodal subgroups of the quotients of compact
classical Lie groups (to appear in J. Lie Theory). CI(n)=Sp(n)/U(n) is the
double covering space of the bottom space denoted by CI(n)^*. In this talk
we classify maximal antipodal sets of CI(n)^*. In order to do that, we use
a certain totally geodesic embedding of CI(n) into Sp(n) and the
classification of maximal antipodal subgroups of Sp(n)/\mathbb{Z}_2.
Moreover, we determine great antipodal sets of CI(n)^* which give the
2-number of CI(n)^*. This talk is based on my joint work with Hiroyuki
Tasaki.
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| Yumiko Kitagawa |
| Title: Duality of singular paths (2,3,5)-distributions
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| Abstract:
We show a duality which arises from distributions of Cartan type,
having growth (2, 3, 5), from the view point of geometric control theory.
In fact we consider the space of singular (or abnormal) paths on a given five
dimensional space endowed with a Cartan distribution, which form
another five dimensional space with a cone structure. We regard the cone
structure as a control system and show that the space of singular paths of the
cone structure is naturally identified with the original space. Moreover we
observe an asymmetry on this duality in terms of singular paths.
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| Naoyuki Koike |
| Title:
Collapse of the regularized mean curvature flow for invariant hypersurfaces in a Hilbert space
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| Abstract:
In this talk, we first state known results for the regularized mean curvature flow starting from
an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group
whose orbits are regularized minimal. Next we prove that, if the initial invariant hypersurface satisfies
a certain kind of horizontally convexity condition, then it collapses to an orbit of the Hilbert Lie group action
along the regularized mean curvature flow. As its application, we state results for the mean curvature flow in
the orbit space of the Hilbert Lie group action, which is a Riemannian orbifold.
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| Yasuyuki Nagatomo |
| Title: Harmonic maps from the complex projective line into complex quadrics
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| Abstract:
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| Hiroshi Tamaru |
| Title:
Quandles and discrete symmetric spaces --- flatness and commutativity
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| Abstract:
The notion of quandles is originated in knot theory,
but they can also be regarded as discrete symmetric spaces.
In this talk, we mention some results on quandles in relation with
symmetric spaces,
mainly on the flatness and commutativity of quandles.
In our sutdies, ideas of the theory of symmetric spaces play important
roles.
We also mention some possibilities of the converse direction, that is,
some of the studies on quandles can possiblly be applied to the studies
on symetric spaces.
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| Yuichiro Taketomi |
| Title: On a Riemannian manifold whose moduli space of invariant metrics is a point
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| Abstract:
In this talk, we introduce a Riemannian metric which always gives a self-similar solution for any “natural” metric evolution equation.
Also, they are a kind of generalization of isotropy irreducible spaces. We gives many examples of such Riemannian metrics.
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| Hyunjin Lee(*) and Young Jin Suh |
| Title: The cyclic parallel hypersurfaces in complex Grassmannians with rank 2
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| Abstract:
In this talk, we introduce a notion of cyclic parallelism for real hypersurfaces in complex Grassmannians of rank two
and give some results related to this notion.
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| Young Jin Suh and Changhwa Woo(*) |
| Title:
The maximal existence condition of real hypersurfaces in complex Grassmannians of rank two
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| Abstract:
In this talk, we introduce a notion of parallel Ricci tensor for Hopf hypersurfaces in complex Grassmannians of rank two.
We use partially ordered class from the the space spaned by the Reeb vector field to the tangent bunddle of real hypersurfac
and try to find the maximal subbundle including Reeb direction which guarantees the existence of real hypersurfaces in
the given ambient spaces.
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| Imsoon Jeong, Gyu Jong Kim(*), and Young Jin Suh |
| Title:
Real hypersurfaces In the Complex quadric with normal Jacobi operator of Codazzi type
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| Abstract:
We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric
$Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The normal Jacobi operator of Codazzi type implies that
the unit normal vector field $N$ becomes $\frak{A}$-principal or $\frak{A}$-isotropic.
Then according to each case, we give a complete classification of real hypersurfaces
in $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with normal Jacobi operator of Codazzi type.
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| Hikaru Yamamoto |
| Title: Ricci-mean curvature flows and its Gauss maps
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| Abstract:
First, I introduce a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow
and a Ricci flow. The ambient metric is evolving under the Ricci flow and
a submanifold is moving in this ambient space along the mean curvature flow.
Recently, Ricci-mean curvature flows have been appeared in some contexts.
In this talk, I will give a generalization of a theorem due to E. Ruh and J. Vilms.
They proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map.
Then, our generalization is a time dependent version of that theorem.
It says that the Gauss maps of a Ricci-mean curvature flow is a vertically harmonic map heat flow.
This is also a generalization of a result due to M.-T. Wang for a mean curvature flow in a Euclidean space.
I will also give some applications of this theorem and its variant.
This talk is based on a joint work with N. Koike.
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| Marina Statha |
| Title: Non-naturally reductive Einstein metrics on compact simple Lie groups
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| Abstract:
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| Isami Koga |
| Title:
Equivariant holomorphic embeddings from the complex projective lines into a complex Grassmannian 2-planes
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| Abstract:
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| Takahiro Hashinaga |
| Title: On homogeneous Lagrangian submanifolds in complex hyperbolic spaces
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| Abstract:
In this talk, we study homogeneous Lagrangian submanifolds in complex hyperbolic spaces.
We show there exists a correspondence between compact homogeneous Lagrangian submanifolds
in complex hyperbolic spaces and ones in complex Euclidean spaces (or equivalently, complex
projective spaces). We also introduce classification results of non-compact homogeneous
Lagrangian submanifolds in complex hyperbolic spaces obtained by actions of connected closed
subgroups of the solvable part of the Iwasawa decomposition. This talk is based on a joint work
with Toru Kajigaya.
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| Toru Kajigaya |
| Title: Reductions of minimal Lagrangian submanifolds with symmetries
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| Abstract:
We give a Hsiang-Lawson type theorem for minimal Lagrangian submanifolds in a Kahler manifold.
More preciously, we show the minimality of a K-invariant Lagrangian submanifold L in a Fano manifold M
w.r.t. a globally conformal Kahler metric is equivalent to the minimality of the reduced Lagrangian submanifold L_0=L/K
in a Kahler quotient w.r.t. the Hsiang-Lawson metric. Moreover, we give some examples of Kahler reductions
by using a circle action obtained from a cohomogenenity one action on a Kahler-Einstein manifold of positive Ricci curvature.
These examples are closely related to homogeneous hypersurfaces with isometric Reeb flows. Applying these results,
we give several examples of minimal Lagrangian submanifolds via reductions.
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| Shinji Ohno |
| Title:
Biharmonic orbits of isotropy representations of symmetric spaces
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| Abstract:
In this talk, we construct biharmonic submanifolds in hyperspheres as orbits of linear isotropy representations
of Riemannian symmetric spaces.
In particular, we obtain examples of biharmonic submanifolds in hyperspheres whose co-dimension is greater than one.
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| Yoshihiro Ohnita |
| Title: On Floer homology of the Gauss images of isoparametric hypersurfaces |
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| Abstract:
Recently we used the Floer homology and the lifted Floer homology for
monotone Lagrangian submanifolds in order to study their Hamiltonian non-displaceability
(H. Iriyeh, H. Ma, R. Miyaoka and Y. Ohnita, Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces,
Bull. London Math. Soc. (2016) 48 (5): 802-812).
In this talk, I would like to explain the spectral sequences for the Floer homology and the lifted Floer homology of
monotone Lagrangian submanifolds and their applications to the Gauss images of isoparametric hypersurfaces,
which are the main technical part in our joint work.
Moreover I will suggest some related open problems for the further research.
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