Naoyuki Koike Laboratory
 

Reserch Thema :

 
Differential Geometry and

 
Geometric Analysis

Welcome to Naoyuki Koike Laboratory

Members of Koike laboratory study Differential Geometry which closely related to Einstein's General Relativity.
Mainly we study Submanifold Theory in Symmetric Spaces.

Main two methods of the research are as follows:

1. Research of submanifolds in symmetric spaces of compact type
   〜 by reducing research of submanifolds in an infinite dimensional
     linear space obtained by canceling holonom of the symmetric space〜
We reasearch submanifolds in a symmetric space
of compact type by reducing to reasearch of submanifolds in an infinite dimensional Hilbert space through a Riemannian submersion of the Hilbert space onto the symmetric space. This reducement of research is valid because symmetric spaces of compact type has holonom but the Hilbert space has no holonom.

2. Research of submanifolds in symmetric spaces of noncompact type
  〜 by reducing to research of submanifolds in an infinite dimensional
    complex linear space obtained by canceling holonom of the complexi-
    fication of the symmetric space〜
We reasearch submanifolds in a symmetric space of compact type by reducing to reasearch of submanifolds in an infinite dimensional anti-Kaehlerian space through a Riemannian submersion of the anti-Kaehlerian space onto the complexification of the symmetric space. This reducement of research is valid in the following two aspects:
(1) A symmetric space of noncompact type has holonom but the infinite dimensional anti-Kaehlerian space has no holonom.
(2) Some geometrical structures (of the submanifold) vanishing beyond the ideal boundary of a symmetric space of non-compact type appear in the complexified space.

3.  Research of the mean curvature flow and the inverse mean curvature flow in    symmetric spaces
We reaserch how a given submanifold collapses (or blows up) along the mean curvature flow and the invesrse mean curvature flow. Also, we research whether a given submanifold converges to a good submanifold along the suitably rescaled mean curvature flow and the suitably rescaled inverse mean curvature flow.

The files used in the intensive lectures at Kyungpook National University (Korea)(18〜20 December, 2013)
Lecture 1 : Click here

Lecture 2 : Click here
Lecture 3 :Click here
Lecture 4 : Click here
Lecture 5 : Click here
(The date of up load:24 January, 2014)


Photos of Cats and Bears


Announcement(April,2017-March,2018)


・ I talked about ``Mean curvature flow for pinched submanifolds in rank one
 symmetric spaces'' at
The 64-th Geometry Symposium
 Date:August 29
 Place:Kanazawa University


・ Conference ``Geometric flows and related problems''
 plan to be held.
 Date:November 23-25
 Place:Tokyo Institute of Technology, Ookayama Campus
 See the following site in detail:

 
http://www.rs.kagu.tus.ac.jp/~koike/gflow2017-e.html

Announcement(April,2016-March,2017)


・ I talked about ``Volume-preserving mean curvature flow for tubes in rank one symmetric
 spaces of non-compact type'' at
 Conference at RIMS ``Regularity, singularity and long time behavior
 for partial differential equations with conservation law''
 Date:June 7
 Place:Research Institute for Mathematical Sciences Kyoto University
 

・I talked about ``Volume-preserving mean curvature flow for tubes in rank one
 symmetric spaces of non-compact type''
at

  ``The 20th International Workshop on Hermitian Symmetric Spaces and
   Submanifolds (and The 12th RIRCM-OCAMI Joint Differential Geometry
   Workshop)''.
 Date : July, 29
 Place:Kyungpook National University, Daegu, Korea

・ Conference ``Submanifold Geometry and Lie Group Action 2016''
 was held.
 Date:September 1-2
 Place:Tokyo University of Science, Morito Memorial Hall
 See the following site in detail:

 
http://www.rs.kagu.tus.ac.jp/~koike/sgla2016.html

・I talked about ``Volume-preserving mean curvature flow for tubes with boundary in symmetric spaces'' at
  ``The First Japan-Taiwan Joint Conference on Differential Geometry (and The 8th TIMS-OCAMI-OCAMI-Waseda Joint International Workshop on Differential Geometry and Geometric Analysis)''.
 Date : December, 13-17
 Place:Waseda University, Japan


Announcement(April,2015-March,2016)


・ The 62-th Geometry Symposium was held.
 Date:August 27-30
 Place:Tokyo University of Science, Kagurazaka Campus
 See the following site in detail:


http://www.rs.kagu.tus.ac.jp/~koike/Geometry-Symposium-2015.html

・ Conference ``Submanifold Geometry and Lie Group Action 2015''
 was held.
 Date:September 7-8
 Place:Tokyo University of Science, Morito Memorial Hall
 See the following site in detail:

 
http://www.rs.kagu.tus.ac.jp/~koike/sgla2015.html

・Conference
 ``Geometric Analysis in Geometry and Topology 2015''
 plans to be held.
 Date:November 9-12
 Place:Tokyo University of Science, Morito Memorial Hall
 See the following site in detail:
 
http://www.rs.kagu.tus.ac.jp/~koike/gagt2015.html 

・Conference
 ``Geometric flows and related problems'' plans to be held.
 Date:March 3-4, 2016
 Place:Tokyo Institute of Technology (Ookayama campus)
 See the following site in detail:
 
http://www.rs.kagu.tus.ac.jp/~koike/gflow.html 

Announcement(2014)


・I talked about ``the regularized mean curvature flow in the Hilbert space'' in

  ``2014 ICM Satellite Cinference on Real and Complex Submanifolds''.
  Date : August, 10-12
  Place:NIMS, Daejeon, Korea
See the following site about the detail:
http://www.icm2014.org/en/program/satellite/satellites

・Conference ``Submanifold Geometry and Lie Group Action 2014''
 was held.
 Date:September 5-6
 Place:Tokyo University of Science, Morito Memorial Hall
 See the following site in detail:

 
http://www.rs.kagu.tus.ac.jp/~koike/sgla2014-e.html

・Conference
``Geometric Analysis in Geometry and Topology 2014''
 was held.
  Date:october 28-31
  Place:Tokyo University of Science, Morito Memorial Hall
 See the follwong site in detail:
 
http://www.rs.kagu.tus.ac.jp/~koike/gagt2014.html




Links


Hobby