To Japanese version
                                                             Kagurazaka Differential Geometry Seminar
              
Record of 2021 April - 2022 March


Date : 29 May
Place : Online by Zoom
15:00-16:00
Speaker : Toru Kajigaya (Tokyo University of Science)
Title : Equivariant realizations of Hermitian symmetric space of noncompact type

Abstract :

Let M=G/K be a Hermitian symmetric space of noncompact type. We provide a way of constructing K-equivariant embeddings from M to its tangent space at the origin by using the polarity of the K-action. As an application, we reconstruct theK-equivariant holomorphic embedding so called the Harish-Chandra realization and the K-equivariant symplectomorphism constructed by Di Scala-Loi and Roos under appropriate identifications of spaces. Moreover, we characterize the holomorphic/symplectic embedding of M by means of the polarity of the K-action. Furthermore, we show a special class of totally geodesic submanifolds in M is realized as either linear subspaces or bounded domains of linear subspaces in the tangent spaceby the K-equivariant embeddings. We also construct a K-equivariant holomorphic symplectic embedding of an open dense subset of the compact dual into its tangent space at the origin as a dual of the holomorphic/symplectic embedding of M. This talk is based on a joint work with Takahiro Hashinaga.
16:15-17:15
Speaker : Shunsuke Saito (Tokyo University of Science)
Title : On polytopeal sufficient conditions for uniform relative K-polystability
    and relative K-instability of polarized toric varieties

Abstract :

It is well known that there is a one-to-one correspondence between polarized toric varieties and integral convex polytopes. Under this correspondence, uniform relative K-polystability/relative K-instability of varieties can be completely described by polytopes and convex functions. In this talk, I will introduce a condition of polytopes that lead to uniform relative K-polystability. I will also explain a result which is related to the polytopal sufficient condition for relative K-instability on toric Fano varieties proved by Yotsutani-Zhou. 

The contents of the talk are based on a joint work with Yasufumi Nitta (Tokyo University of Science).


Date : 24 July
Place : Online by Zoom  
15:00-16:00
Speaker : Shota Hamanaka (Chuo University)
Title : Closed Ricci flow with integral or pointwise bound of the scalar curvature

Abstract :
 In this talk, we will discuss extension problem on the Ricci flow with bounded some sort of geometric quantity.
 In 1982, Hamilton introduced the Ricci flow which deforms Riemannian metrics in the direction of the Ricci tensor.One hopes that the Ricci flow will deform any Riemannian metric to some canonical metrics. He also proved that the Ricci flow equaiton on
a closed manifold has a unique short time solution for any initial metric. The next immediate question is the so-called ``maximal existence time'' for the Ricci flow with respect to initial metric. Hamilton proved that T < +∞ is the maximal existence time of a closed Ricci flow
(M, g(t)) (t ∊ [0,T), dim M ≧ 2) if and only if its norm of the Riemannian curvature tensor is unbounded as $t \rightarrow T$. Therefore a uniform bound for the norm of the Riemannian curvature tensor on M × [0,T) is enough to extend the Ricci flow smoothly over T. Moreover, another known such sufficient conditions to extend the flow smoothly are,
for example,
uniformly bound of the norm of the Ricci curvature tensor on M × [0,T) (by Sesum) or

certain integral bounds of geometric quantities rather than point-wise one (by Wang). 
Di-Matteo generalized Wang's results using mixed integral norms (i.e., space and time)
which is parametrized by (α, β) ∊ (1, ∞).

 
In the talk, we will give new extendable result on the closed four dimensional Ricci flow
under the conditions corresponding to (α, β) = (p, +∞) (p > 2)
and (+∞, 1). If there is
time, we also discuss the condition sup_{|R| | x ∊ M × [0,T)}  < +∞ (the uniformly bound of
the scalar curvature).

16:15-17:15
Speaker : Youhei Sakurai (Saitama University)
Title : Liouville theorems for harmonic map heat flow
    along ancient super Ricci flow via reduced geometry
Abstract :

This talk is based on the joint work with Keita Kunikawa (Utsunomiya university). We have obtained a Liouville theorem for heat equation along ancient super Ricci flow via Perelman’s reduced geometry. Recently, we generalize the target spaces, and formulate several Liouville theorems for harmonic map heat flow. In this talk, I will introduce such Liouville theorems. In our Liouville theorems, we impose a growth condition concerning Perelman’s reduced distance. For non-positively curved target spaces, our growth condition is sharp. On the other hand, for positively curved target spaces, it is not clear that our growth condition is sharp. However, in the static case, the growth condition can be improved, which is almost sharp in view of the example constructed by Schoen-Uhlenbeck. I will discuss the sharpness in more detail in the talk.


Date : 28 August
Place : Online by Zoom
15:00-16:00
Speaker : Tomoki Fujii (Tokyo Uiniversity of Science)
Title : Graphical translating solitons for the mean curvature flow and isoparametric functions

Abstract :

In this talk, I will discuss the condition that the graph of a function over a Riemannian
manifold M whose level sets give isoparametric foliation is a translating soliton for the
mean curvature flow.This translating soliton is given as the graph of a function which
is given as a composition of
an isoparametric function over M and a solution of a certain
ordinary differential equation. I will introduce this in detail in the talk. In addition, I will
explain the classification of the shape of such translating solitons in case where the
Riemannian manifold M is the n-dimensional sphere (n
2).


16:15-17:15
Speaker : Takuma Tomihisa (Waseda University)
Title :The Rarita-Schwinger operator and some related topics
Abstract:

 The Rarita-Schwinger fields play an important role in physics by describing the gravitino. 

The Rarita-Schwinger operator is also important to define the Rarita-Schwinger fields. 

Recently, in mathematics, there are many researches about the Rarita-Schwinger operator,
the "spin-3/2 version" of the Dirac operator
. In this talk, we state some researches about

(1) spectra of the Rarita-Schwinger operators on some symmetric spaces (joint work with
Yasushi Homma), 

(2) the Rarita-Schwinger fields on nearly Kähler manifolds (joint work with Soma Ohno).



  Organizers : Naoyuki Koike, Makiko Tanaka,
Akifumi Sako,
         Yasufumi Nitta,
Kurando Baba, Toru Kajigaya,
         Shunsuke Saito,Tsukasa Takeuchi

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