| Title and Abstract of Talks：
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| Xiaohuan Mo (This talk was cancelled.) |
| Title: Finsler warped product metrics
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| Abstract:
In this lecture we discuss the warped structures of Finsler metrics. We obtain the differential equation that characterizes the Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics. We also refine and improve Chen-Shen-Zhao equations characterizing Finsler warped product metrics of scalar flag curvature. In particular, we find equations that characterize Finsler warped product metrics of constant flag curvature. Then we improve Chen-Shen-Zhao result on characterizing Einstein Finsler warped product metrics. As its application we construct explicitly many new warped product Douglas metrics of constant Ricci curvature by using known locally projectively flat spherically symmetric metrics of constant flag curvature.
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| Young Jin Suh (This talk was cancelled.) |
| Title:
Differential geometry of complex hyperbolic quadrics and related topics |
| Abstract:
In this talk, we want to explain geometric structures of complex hyperbolic quadrics
$Q^m^*=SO_{2,m}/SO_2{\times}SO_m$ which was recently developed by Klein and Suh ([5]).
Related to this structure, we give some research topics on real hypersurfaces in the complex
hyperbolic quadric with parallel Ricci tensor, harmonic curvature or pseudo-anti commuting Ricci tensor
([1], [2],[3],[4]).
References:
[1] Y.J. Suh, Real hypersurfaces in the complex hyperbolic quadric with isometric Reeb flow,
Commun. Contemp. Math. 15(2018), 1750031(20 pages).
[2] Y.J. Suh, Pseudo-anti commuting Ricci tensor for real hypersurfaces in the complex hyperbolic quadric,
Vol. 62 No. 4(2019), 679-698.
[3] Y.J. Suh, Real hypersurfaces in the complex hyperbolic quadric with harmonic curvature, to appear in Math. Nachr. (2020).
[4] G.J. Kim and Y.J. Suh, Real hypersurfaces in the complex hyperbolic quadric with parallel Ricci tensor, Results in Math., 74(2019), 33(30 pages).
[5] S.Klein and Y.J.Suh, Contact real hypersurfaces in the complex hyperbolic quadric,
Ann. Mat. Pura Appl. 198(2019), 1481-1494.
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| Byung Hak Kim (This talk was cancelled.) |
| Title: Ricci and Yamabe solitons with related topics
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| Abstract:
Recently, there has been a lot of research on slitons and a lot of the results has been published. In this talk we summarize and report of the recent results of the (almost) Ricci and Yamabe solitons on the warped product of two Riemannian manifolds. We are going to talk about the classification of the manifold with (almost)Ricci or Yamabe solitons and the construction of the model space. Finally we want to think about the open problems related to this topics.
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| Changhwa Woo*,
Oscar J. Garay(University of the Basque Country UPV/EHU, Spain) (This talk was cancelled.) |
| Title: Extremal Hypersurfaces constrained elastica in Lorentzian space forms |
| Abstract:
We study geodesics in hypersurfaces of a Lorentzian space,
which are critical curves of the bending energy functional, for variations
constrained to lie on the hypersurface. We characterize critical geodesics
showing that they live fully immersed in a totally geodesic and that they
must be of three different types. Finally, we consider the classification of
surfaces in the Minkowski 3-space foliated by critical geodesics.
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| Imsoon Jeong (This talk was cancelled.) |
| Title:
Real hypersurfaces in the complex quadric with Killing structure Jacobi operator |
| Abstract:
In this talk, we introduce the notion of Killing structure Jacobi operator for real hypersurfaces in the complex quadric. Next we give a complete classification of real hypersurfaces in the complex quadric with Killing structure Jacobi operator.
References:
[1] J. Berndt, Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, Int. J. Math. 24 (2013) 1350050 (18 pages).
[2] D. E. Blair, Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (1971) 285–292.
[3] S. Klein, Totally geodesic submanifolds in the complex quadric, Differential Geom. Appl. 26 (2008) 79–96.
[4] I. Jeong, J. D. Pérez and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannian with parallel structure Jacobi operator, Acta. Math. Hungar. 122(1-2) (2009) 173–186.
[5] Y. J. Suh, Real hypersurfaces in the complex quadric with parallel structure Jacobi operator, Differential Geom. Appl. 51 (2017) 33–48.
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| Hyunjin Lee* & Young Jin Suh (This talk was cancelled.) |
| Title:
Cyclic parallelism of real hypersurfaces in the real Grassmannians with rank 2 |
| Abstract:
In this talk, we will give some characterization for Hopf real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The complex quadric $Q^{m}$ is a kind of real Grassmannians with rank 2 of compact type, which is given by a complex hypersurface in the complex projective space. Accordingly, $Q^{m}$ admits both a real structure and a complex structure with anti-commutes with each other. From these geometric structures of $Q^{m}$ we can also induce the geometric structures of a real hypersurface $M$ in $Q^{m}$. By using these structures, in this talk we will give some classification problems for Hopf real hypersurfaces in $Q^{m}$ in terms of cyclic parallelism with respect to (1,1) type tensors of $M$.
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| Naoyuki Koike |
| Title:
Calabi-Yau structures and Special Lagrangian submanifolds of complexified symmetric spaces
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| Abstract:
In 1993, M. Stenzel constructed complete Ricci-flat Kaehler metrics on the complexificaion of
rank one symmetric spaces of compact type, where the complexificaion of the symmetriec space is defined as
the tangent bundle of the symmetric space equipped with the adapted complex structure.
The complete Ricci-flat Kaehler metrics is called the "Stenzel metric".
In 2004, R. Bielawski gave a construction of (not necessarily complete) Ricci-flat Kaehler metrics on
the complexifications of general rank symmetric spaces of compact type, where the complexificaion of the symmetric space also is as above.
In this talk, we first state a new explicit construction of complete Ricci-flat Kaehler metrics
on the complexifications of general rank symmetric spaces of compact type.
This construction is given by using the Schwarz's theorem on the basis of the above Bielawski's construction.
This complete Ricci-flat metric together with the natural complex structure and the natural holomorphic $(n,0)$-form on the complexification gives the Calabi-Yau structure on the complexification.
Symmetric subgroups of the isometry group of the symmetric space act on the complexification of the symmetric space as Hamiltonian actions. Next we state constructions of special Lagrangian submanifolds of any phase,
which are invariant under the symmetric subgroup action, in the complexification of the symmetric space equipped with the above Calabi-Yau structure.
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| Sachiko Hamano |
| Title: On variational formulas for hydrodynamic differentials and its application
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| Abstract:
We investigate conformal embeddings of a marked open Riemann surface of finite genus into closed Riemann surfaces with the same genus. The hydrodynamic differentials play an important role in describing the moduli space of conformal embeddings. From the viewpoint of several complex variables, we establish the variational formulas of hydrodynamic differentials for a family of open Riemann surfaces of finite genus with a complex parameter. We utilize these formulas in characterizing the pseudoconvexity of the deformation space.
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| Yasuyuki Nagatomo |
| Title: Harmonic mappings into Grassmannians
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| Abstract:
We generalize Theorem of Takahashi on minimal isometric immersions into a sphere using a geometry of vector bundles
and apply it to obtain a generalization of the theory of do Carmo and Wallach.
A generalized do Carmo-Wallach theory gives a moduli space of harmonic maps
of compact Riemannian manifolds
into Grassmannians.
Moreover the moduli space is equipped with the topology induced by $L^2$-scalar product on
the space of sections of a vector bundle and we explain the geometric meaning of the
compactification of the moduli space.
In addition, for each holomorphic map of a K\"ahler manifold into complex Grassmannian with the positivity condition,
we give a holomorphic map into complex projective space called an associated map.
The domain of the associated map is {\it different} from the domain of the original holomorphic map.
As applications of the compactification and associated maps,
the moduli spaces of holomorphic isometric embeddings of the complex projective line
into a complex hyperquadric are described with the help of representation theory of Lie groups.
We use the compactification of the moduli spaces to classify equivariant harmonic maps of the complex projective spaces
into complex Grassmannians of low rank.
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| Takashi Sakai |
| Title: Natural Γ-symmetric structures on R-spaces
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| Abstract:
The notion of Γ-symmetric spaces was introduced by Lutz in 1981 generalizing k-symmetric spaces. In this talk, we consider Γ-symmetric structures on some R-spaces using Γ-symmetric triples introduced by Goze and Remm. We give a characterization in terms of root systems and classification of R-spaces that admit a certain natural Γ-symmetric structure, where Γ is a power of Z_2. For Γ=Z_2, we recover the symmetric R-spaces.
In 1988, Chen and Nagano studied antipodal sets of compact symmetric spaces. The notion of antipodal sets can be extended to Γ-symmetric spaces. We study antipodal sets of R-spaces with respect to their natural Γ-symmetric structures, and show that a maximal antipodal set is given as an orbit of the Weyl group. That is a generalization of a result on maximal antipodal sets of symmetric R-spaces due to Tanaka and Tasaki.
This talk is based on a joint work with Peter Quast.
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| Nobutaka Boumuki |
| Title: Paraholomorphic cohomology groups of hyperbolic adjoint orbits
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| Abstract:
This talk is based on a joint work with Tomonori Noda (Meiji Pharmaceutical University).
For a paracomplex manifold, we construct certain cohomology groups. Then we clarify a link between such cohomology groups of hyperbolic adjoint orbits and the de Rham cohomology groups of real flag manifolds. That establishes relation between a paraholomorphic invariant and a topological invariant.
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| Takayuki Okuda |
| Title: Totally geodesic immersions of direct products of 2-spheres in compact symmetric spaces
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| Abstract:
Let M be a connected compact symmetric space. The rank of M can be understand as the maximum dimension of tori (direct products of copies of 1-sphere) which admit totally geodesic immersions into M. In this talk, we study the maximum number N for each M such that the direct products of N-copies of 2-spheres admits a totally geodesic immersions into M. We also discuss an applications of such the number N for proper actions on some pseudo-Riemannian symmetric spaces.
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| Toru Kagigaya |
| Title: Uniformizing surfaces via discrete harmonic maps
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| Abstract:
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface,
there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among
a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces
as a refinement of the Nielsen realization problem for the mapping class groups. It is a joint work with Ryokichi Tanaka (Tohoku).
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| Keita Kunikawa |
| Title: On Ecker's integral at infinity on ancient mean curvature flows
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We show that Ecker's local integral quantity and Huisken's global integral quantity agree at infinity on ancient mean curvature flows. As an application, we obtain Colding-Minicozzi’s codimension bound by the entropy.
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| Akifumi Ochiai |
| Title: A construction of Lagrangian mean curvature flows by generalized perpendicular symmetries
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We show a method to construct a Lagrangian mean curvature flow from a given special Lagrangian submanifold in a Calabi-Yau manifold by generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. We construct some examples in C^n by our method.
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| Masahiro Morimoto |
| Title: Minimal PF submanifolds in Hilbert spaces with symmetries
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In finite dimensional Riemannian manifolds there are special kinds of minimal submanifolds with certain symmetries, namely reflective submanifolds, weakly reflective submanifolds, austere submanifolds and arid submanifolds. In this talk we introduce the concepts of these submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that in Hilbert spaces there exist many infinite dimensional minimal PF submanifolds with such symmetries. In particular each fiber of the parallel transport map is shown to be a weakly reflective PF submanifold. Those imply that in infinite dimensional Hilbert spaces there exist many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.
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| Yuichiro Sato |
| Title: Totally umbilical submanifolds in pseudo-Riemannian space forms
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Totally umbilical submanifolds in pseudo-Riemannian manifolds are defined by the traceless part of the second fundamental form vanishing identically. In this talk, we classify the congruent class of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms. As applications, we consider moduli spaces of isometric immersions between space forms and non-zero mean curvature parallel submanifolds in pseudo-spheres.
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| Luis Pedro Castellanos Moscoso |
| Title: A classification of left-invariant symplectic structures on some Lie groups
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In geometry it is an important problem to study whether a given manifold admits some nice geometric structures. In the setting of Lie groups is natural to ask about the existence of left-invariant structures. A \textbf{symplectic Lie group} is a Lie group $G$ endowed with a left-invariant symplectic form $\omega$ (i.e. a nondegenerate closed 2-form). There are many interesting results about the structure of symplectic Lie groups and considerable classification efforts in low dimensions, but the general picture is far from complete. In this talk we develop a method to classify left-invariant structures on a given Lie group: we first study the "moduli space of left-invariant nondegenerate 2-forms" and then we search inside this moduli space for the 2-forms that are symplectic. Finally we apply the method to some concrete examples.
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