Updated on 2022/03/31

KOBAYASHI, Ryoichi

Organization
Graduate School of Mathematics Division of Mathematics Fundamental Mathematics Professor
Graduate School
Graduate School of Mathematics
Undergraduate School
School of Science
Contact information

### Degree 1

1. Doctor of Science （ 1987.2   The University of Tokyo ）

### Research Interests 2

1. Complex geometry : Value distribution of holomorphic curves in complex manifolds and its application to Diophantine geometry and minimal surfaces.

2. Differential Geometry : Measure theoretic approach to value distribution of holomorphic curves in algebraic varieties

### Research Areas 1

1. Others / Others  / Geometry

### Current Research Project and SDGs 4

1. Calabi-Yau singular perturbation and its geometric applications

2. Nevanlinna-Galois Theory

3. holomorphic curves in compact complex parallelizable manifolds

4. Hamiltonian volume minimizing property of Hamiltonian minimal Lagrangian submanifolds in compact Kahler manifolds

### Research History 1

1. Professor, Graduate School of Mathematics, Nagoya University

1995.4

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Country：Japan

### Education 2

1. Osaka University   Graduate School, Division of Natural Science   Department of Mathematics

1980.4 - 1982.2

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Country： Japan

2. The University of Tokyo   Faculty of Science   Department of Mathematics

1976.4 - 1980.3

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Country： Japan

### Professional Memberships 1

1. Mathematical Society of Japan

### Awards 1

1. Geometry Prize

1994   Geometry Branch, Mathematical Society of Japan

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Country：Japan

### Papers 12

1. Problems related to Kobayashi hyperbolicity -- A proposal of asymptotic methods Invited

Ryoichi Kobayashi

RIMS Kokyuroku   Vol. 2211   page： 153 - 164   2022.1

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Authorship：Corresponding author   Language：English   Publishing type：Research paper (conference, symposium, etc.)

2. Logarithmic Sobolev inequalities, entropy formula and Riemannian analogue of thermistat -- Perelman's approach to Ricci flow, I -- Invited Reviewed

Ryoichi Kobayashi

Sugaku Exposition   Vol. 60 ( 3 ) page： 225-247   2008

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Authorship：Lead author   Language：Japanese

Perelman completed Hamilton's program by proving his local
noncollapsing theorem for the Ricci flow and proved the geometrization conjecture. In this survey paper, I provided
a comprehensible exposition on Perelman's local noncollapsing theorem with its rich mathematical background
as well as ideas from statistical physics. I put emphasis
on Perelman's orogonal ideas indicated in the title of
this paper, which are not fully exposed in other survey papers, and I explained an intuitive physical meaning of
Perelman's local noncollapsing theorem from these view
points.

3. Logarithmic Sobolev inequalities, entropy formula, Riemannian geometric thermostat -- Perelman's approach to the Ricci flow, II -- Invited Reviewed

Ryoichi Kobayashi

SUGAKU (Iwanami)   Vol. 60 ( 4 ) page： 352-367   2008

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Authorship：Lead author   Language：Japanese

4. * The Gauss map of pseudo-algebraic minimal surfaces Reviewed

Yu Kawakami, Ryoichi Kobayashi, Reiko Miyaoka

Forum Mathematicum   Vol. 20 ( 6 ) page： 1055-1069   2008.11

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)

Pseudo-algebraic minimal surfaces satisfy all conditions of
algebraic minimal surfraces except period condition. We
generalized Osserman's algebro-geometric argument for a
wider class of pseudo-algebraic minimal surfaces and
proved the best possible estimate for the TRVN of the
Gauss map. The advantage of our generalization is that
we can construct its Nevanlinna analogue (though far from
automatic) .

5. Toward Nevanlinna Theory as a geometric model of Diophantine approximation Invited Reviewed

Ryoichi Kobayashi

SUGAKU EXPOSITION   Vol. 16   page： 39-79   2003

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(Diophantine / Nevanlinna) analogy is most impressive in
(Roth / Nevanlinna's 2nd Main) Theorems. However, the
origin of this similarity is far from being understood.
I attempted to reconstruct Nevanlinna Theory so that everything is reduced to the geometrized LLD so that one
can avoid sophisticated differentiation technique which is hard to tranlate in Diophantine setting. I proposed a
geometric LLD as a defining equation" of derivatives of
rational points in Diophantine setting. This attempt is
based on this proposal.

6. An attempt toward Diophantine analogue of ramification counting Nevanlinna Theory : Truncated counting function in Schmidt's Subspace Theorem (Preliminary Version) Invited

Ryoichi Kobayashi

RIMS koukyuroku   Vol. 1451（代数的整数論とその周辺）   page： 72-111   2005.10

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A geometry which unifies Roth theorem and Nevanlinna's
2nd main theorem still does not exist in satisfactory
form. This paper is based on author's attempt toward
constructing such geometry. I defined a Diophantine
analogue of the ramification counting function by
using the Diophantine analogue of geometrized LLD in
Nevanlinna Theory as the defining equation" of the
jets of rational points". Then, at least formally,
the ramification counting function appears in Schmidt's
subspace theorem and this implies the abc conjecture
(therefore we get a formal evidence of the abc
conjecture).

7. * Toward Nevanlinna-Galois theory for algebraic minimal surfaces Invited Reviewed

Proc. of the 16th OCU International Symposium 2008 OCAMI Studies   Vol. Volume 3   page： 129-136   2010.4

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I proposed a Nevanlinna theory coupled with the Galois group action
and its application to the value distribution of the Gauss map of
algebraic minimal surfaces.

8. * Toward Nevanlinna-Galois theory for algebraic minimal surfaces Invited Reviewed

Proc. of the 16th OCU International Symposium 2008 OCAMI Studies   Vol. Volume 3   page： 129-136   2010

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)

I proposed a Nevanlinna theory coupled with the Galois group action
and its application to the value distribution of the Gauss map of
algebraic minimal surfaces.

9. Nevanlinna-Galois theory for algebraic minimal surfaces Invited Reviewed

Ryoichi Kobayashi

OCAMI Studies, Riemann Surfaces, Harmonic Maps and Visualization   Vol. 3   page： 129-136   2009.2

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10. * Holomorphic curves in Abelian varieties : the second main theorem and applications Reviewed

Ryoichi Kobayashi

Japanese Journal of Mathematics   Vol. 26   page： 129-152   2000

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11. * Minimizing currents is open manifolds and the n-1 homology of nonnegatively Ricci curved manifolds Reviewed

Yoe Itokawa and Ryoichi Kobayashi

American Journal of Mathematics   Vol. 121   1999

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12. * Ricci-flat Kaehler metrics on affine algebraic manifolds and degeneration of Kaehler-Einstein K3 surfaces Invited Reviewed

Ryoichi Kobayashi

Adv. Stud. Pure Math   Vol. 18 ( II ) page： 137-228   1990

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)

### Books 4

1. 朝倉書店「幾何解析」第５章．リッチフローと複素幾何．

小林亮一（ Role： Joint author）

朝倉書店  2018

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Language：Japanese Book type：Scholarly book

Perelmanによる幾何化予想解決が複素幾何にもたらした大きな影響について解説していくつかの問題提起を行った．

2. リッチフローと幾何化予想

小林亮一（ Role： Sole author）

培風館  2011.6  （ ISBN:978-4-563-00665-5

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Language：Japanese

1970年代後半にサーストンによって予想された3次元閉多様体
の幾何化が，ハミルトンとペレルマンによっていかに解決され
たかのか，解決にいたる議論の全容を述べた本である．

3. Surveys in Geometry, special edition

H. Nakajima (ed.)（ Role： Joint author）

2005

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Language：Japanese

I restructured the fundamentals of Nevanlinna Theory
on the fotting of Geometrized Lemma on Logarithmic
Derivative. The idea is that the Nevanlinna Theory is
an intersection theory in partially infinite dimensions.
By taking the average over the finite dimensional part
(i.e., moving ample divisors) I established a geometric
LLD. This open a way toward various geometric applications
of Nevanlinna Theory.

4. Mathematics in 21st century -- Open problems in Geometry --

R. Miyaoka / M. Kotani (ed.)（ Role： Joint author）

2004

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Language：Japanese

Collection of problems in geometry with flavour of
asymptotic analysis. These problems are chosen from
Diophantine/Nevanlinna analogy, Hitchin-Kobayashi
correspondence, Geometric Quantization, Mirror Conjecture,
Gauss map of minimal surfaces, etc.

### Presentations 46

1. Probabilistic Riemann-Hurwitz Formula and Measure Concentration Phenomenon – How to Define the Wronskian – Invited International conference

Ryoichi Kobayashi

2021 GeoQuant  2021.8  Freiburg University

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Event date： 2021.8

Language：English   Presentation type：Oral presentation (invited, special)

Venue：Freiburg   Country：Germany

2. Problems related to Kobayashi hyperbolicity -- A proposal of asymptotic methods -- Invited

2021.9.8

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Event date： 2021.9

Language：Japanese   Presentation type：Oral presentation (invited, special)

Venue：online   Country：Japan

3. Rigidity of compact holomorphic curves in compact complex parallelizable manifold $\Gamma\backslash SL(2,C)$ Invited

Ryoichi Kobayashi

24th Complex Geometry Symposium

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Event date： 2018.11

Language：English   Presentation type：Oral presentation (general)

Venue：Kanazawa University   Country：Japan

I gave a talk on the following result : Any non-constant holomorphic map from a compact Riemann surface $M$ into a compact complex parallelizable manifold $X=\Gamma\backslash SL(2,C)$ decomposes into the Albanese map $a:M \rightarrow Alb(M)$, a homomorphism $h:Alb(M) \rightarrow T$ ($T$ being a maximal torus in $X$) and a right translation
$t:X\rightarrow X$ be an element of $SL(2,C)$. Applications to the deformation theory of compact complex parallelizable manifolds and the CMC surfaces in the hyperbolic spaces are given.

4. Holomorphic Curves in Compact Complex Parallelizable Manifold $\Gamma\backslash SL(2;C)$ International conference

Ryoichi Kobayashi

The 5th Workshop Complex Geometry and Lie Groups

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Event date： 2018.6

Language：English   Presentation type：Oral presentation (general)

Venue：Firenze University   Country：Italy

I gave a talk on my recent result on the rigidity of compact holomorphic curves into a compact complex parallelizable manifold $X$ uniformized by $SL(2,C)$. Any holomorphic map from a compact Riemann surface $M$ into $X$ is a composition of the Albanese $M$ to $Alb(M)$, a group homomorphism of $Alb(M)$ to a maximal torus $T$ in $X$ and a right translation.

5. Holomorphic curves in compact cpmplex parallelizable manifolds \Gamma\backslash SL(2,C) Invited International conference

Ryoichi Kobayashi

The 5th workshop "Complex Geometry and Lie Groups"

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Event date： 2018.6

Language：English   Presentation type：Oral presentation (general)

Venue：Firenze University   Country：Italy

I gave a talk on my recent result on the rigidity of compact holomorphic curves in a compact complex parallelizable manifold $X$ of type $\Gamma\backslash SL(2,C)$. Any holomorphic map from a compact Rirmann surface $M$ into $X$ is expressed as a composition of the Albanere map $X \rightarrow J$ and a homomorphism of $J$ to a maximal torus in $X$.

6. Holomorphic curves in compact complex parallelizable manifolds Invited International conference

Ryoichi Kobayashi

Analytic and Algebraic geometry

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Event date： 2018.3

Language：English   Presentation type：Oral presentation (general)

Venue：ICTS, Bangalore   Country：India

Let $¥Gamma¥subset¥SL(2,¥C)$ be a cocompact lattice and $X=¥Gamma¥backslash¥SL(2,¥C)$
the associated compact complex parallelizable manifold. We show that any holomorphic map $f¥,:¥,M ¥rightarrow X$
from a compact Riemann surface $M$ into a compact complex parallelizable manifold $X$ is expressed
as a composition $f=t¥circ h¥circ ¥alpha$ where $¥alpha¥,:¥,M¥rightarrow ¥Alb(M)$ is the Albanese map,
the the map $h¥,:¥,¥Alb(M) ¥rightarrow X=¥Gamma¥backslash ¥SL(2,¥C)$ has its image in a maximal torus
$T=¥Gamma¥cap A¥backslash A¥cong ¥Z¥backslash ¥C^*$ in $X$ defining an algebraic group homomorphism
$h:¥Alb(M) ¥rightarrow T=A¥cap¥Gamma¥backslash A$,
and $t$ is a right translation by some element of $¥SL(2,¥C)$. We propose two applications of this result :
(i) non-existence of closed immersed CMC-1 surfaces in compact hyperbolic 3-manifold and (ii) the surjectivity
of non-constant holomorphic maps from between small deformations of compact complex parallelizable manifolds
of type $¥Gamma¥backslash ¥SL(2,¥C)$.

7. Holomorphic Curves in Compact Complex Parallelizable Manifold $Γ \backslash SL(2, C)$ Invited International conference

Ryoichi Kobayashi

ANALYTIC AND ALGEBRAIC GEOMETRY (/DISCUSSION-MEETING/AAG2018)

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Event date： 2018.3

Language：English   Presentation type：Oral presentation (general)

Venue：ICTS, Bangalore   Country：India

I gave a lecture on my recent result on the rigidity of compact holomorphic curves in a compact complex parallelizable manifold of type $\Gamma\backslash SL(2,C)$.

8. Quantization of Osserman's Gauss map theory of minimal surfaces Invited

Ryoichi Kobayashi

The 23rd Symposium on Complex Geometry

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Event date： 2017.11

Language：English   Presentation type：Oral presentation (invited, special)

Venue：Kanazawa   Country：Japan

I proposed a partition function which exists behind Osserman's theory. Taking the Planck constant zero limit before summation, I get the basic ratio which explains everything of Osserman's theory. Taking the summation first and then Planck constant zero limit, I encounter an object interpreted as the semi-classical limit of quantized Osserman's theory. I introduced Nevanlinna theory approach to the semi-classical analysis. In particular, I proposed the quantization of the period condition of algebraic minimal surfaces.

9. Quantization of Osserman's Gauss map theory Invited International conference

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Event date： 2017.10 - 2017.11

Language：Japanese   Presentation type：Oral presentation (invited, special)

Venue：Fukuoka University   Country：Japan

10. A Quantization of Osserman's theory of algebraic minimal surfaces International conference

Ryoichi Kobayashi

Geoquant 2017, QGM, Denmark

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Event date： 2017.7 - 2017.8

Language：English   Presentation type：Oral presentation (general)

Venue：Aarhus and Sonderborg, Denmark   Country：Denmark

I propose a quantization scheme of
Osserman's theory on algebraic minimal surfaces.
Starting with an observation that the basic quantity
in Osserman's theory is interpreted as a sort of
partition function in which the limit h to 0 comes
fi rst before summation, I will answer the question
what happens if we perform summation first and
then take the limit h to 0. We arrive at a geometry
completely different from Osserman's.

11. Holomorphic curves in compact complex parallelizable manifolds

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Event date： 2017.7

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

12. Quantization of Osserman's theory on algebraic minimal surfaces Invited

Ryoichi Kobayashi

Hayama SCV symposium 2017

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Event date： 2017.7

Language：English   Presentation type：Oral presentation (invited, special)

Venue：Shonan Village   Country：Japan

I point out that there exists a partition function behind Osserman's theory of algebraic minimal surfaces. The key concept in Osserman's theory is the basic ratio which is obtained by first letting Planck constant to 0 and then summation. I introduced the audience a new mathematics arising from interchanging the limit and the summation into the usual order in the semi-classical limit.

13. 代数的極小曲面のオッサーマン理論の量子化するにあたって現れる種々の問題 Invited

小林亮一

北九州幾何学研究会2017

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Event date： 2017.7

Language：Japanese   Presentation type：Oral presentation (invited, special)

Venue：九州工業大学   Country：Japan

代数的極小曲面のオッサーマン理論の背景に分配関数が在ることを指摘し伝統的なオッサーマン理論ではプランク定数0の極限をとってから和をとる．この順序を準古典極限の順序である，先に和をとってからプランク定数を0に持って行く順序に戻してやるとどんな数学が現れるかについて論じた．

14. Quantization of Osserman's theory on algebraic minimal surfaces Invited

Ryoichi Kobayashi

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Event date： 2017.3

Language：Japanese   Presentation type：Oral presentation (invited, special)

Venue：Meiji University   Country：Japan

There exists a partition function on the fundamental group of an algebraic minimal surface which recovers Osserman's theory as a version first taking letting the limit Planck constant 0. In this lecture I will talk about geometry arising if one exchanges the order of the sum and limit, i.e., sum first and limit.

15. Free Fuchsian groups in classical minimal surface theory International conference

Ryoichi Kobayashi

The 4-th Workshop Complex Geometry and Lie Grouls''

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Event date： 2016.3

Language：English   Presentation type：Oral presentation (general)

Country：Japan

In this lecture, I reveal the prominent role of free Fuchsian groups (which
appears as fundamental groups of complete minimal surfaces with finite
total curvature) in the classical minimal surface theory.

16. Geometry of parabolic localization

Ryoichi Kobayashi

Geometry and Analysis Fukuoka

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Event date： 2015.10 - 2015.11

Language：Japanese   Presentation type：Oral presentation (general)

Country：Japan

In this talk I discussed the localization phenomenon arising from the
action of free Fuchsian groups on the universal covering surface of
complete minimal surfaces with finite total curvature. I showed that
the localization phenomenon plays an essential role in the value distribution of the Gauss map.

17. Legendre duality for general polarization

Ruoichi Kobayashi

21-th Complex Geometry Workshop

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Event date： 2015.10

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

In this taok I introduced an idea of extending the Legendre duality
in the setting of anti-canonical polarization to general polarization.
This idea is useful in the attempt of generalizing the monotonicity of the
Mabuchi functional along the modified Ricci iteration (a natural generalization of the Ricci iteration to general polarization).

18. Metrization of the Osserman Theory International conference

Ryoichi Kobayashi

Distribution of Rational Points and Rational Curves in Algebraic Varieties

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Event date： 2015.3

Language：English   Presentation type：Oral presentation (general)

Country：Canada

I proposed a metrization of the Osserman theory on the Gauss map of
algebraic minimal surfaces and discuss on some new results obtained
by the metrization.

19. Mabuchi Functionals

Ryoichi Kobayashi

Mabuchi65

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Event date： 2015.3

Language：Japanese   Presentation type：Oral presentation (general)

Country：Japan

20. Localization arising from iteration of parabolic translation and collective Cohn-Vossen inequality International conference

Ryoichi Kobayashi

5-th International Conference on Differential Geometry and Analysis

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Event date： 2014.5 - 2014.6

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

21. Relations among various invariants on algebraic minimal surfaces

Ryoichi Kobayashi

The 19-th Symposium on Complex Geometry

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Event date： 2013.10 - 2013.11

Language：English   Presentation type：Oral presentation (general)

Country：Japan

We introduce a Nevanlinna theoretic analogue of the
Kawakami-Kobayashi-Miyaoka invariant of algebraic minimal surfaces
and compare it with various invariants. We try to understand the problem
on the best possible upper bound for the number of exceptional values
og the Gauss map of algebraic minimal surfaces.

22. Hamiltonian Volume Minimizing Property of Maximal Torus Orbits in the Cpmplex Projective Space International conference

Ryoichi Kobayashi

ESI School and Conference Geometry and Quantization GEOQUANT 2013

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Event date： 2013.8

Language：English   Presentation type：Oral presentation (general)

Country：Austria

23. Hamiltonian Volume Minimizing Property of $U(1)^n$-orbits in $CP^n$ International conference

Ryoichi Kobayashi

The 5th OCAMI-TIMS Joint International Conference on Differential Geometry and Geometric Analysis

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Event date： 2013.3

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

24. Hamiltonian Volume Minimizing Property of Maximal Torus Orbits in the Cpmplex Projective Space International conference

Ryoichi KObayashi

Conference on the occasion of Martin Schlichenmaier's 60th Birthday

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Event date： 2012.12

Language：English   Presentation type：Oral presentation (general)

Country：Japan

I proposed an idea for the proof of volume minimizing property of maximal torus orbits under Hamiltonian deformation.

25. Nevanlinna Theory from the View Point of Lemma on Logarithmic Derivative International conference

Ryoichi Kobayashi

Mini-courses and Workshop - The Topology of Algebraic Varieties

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Event date： 2012.9

Language：English   Presentation type：Oral presentation (general)

Country：Canada

BMY inequality is interpreted as a Diophantine inequality. In this lecture,
I proposed a logical structure behind BMY-type inequalities from the view
point of Lemma on Logarithmic Derivative in Nevanlinna Theory.

26. Hamiltonian K¥"ahler-Ricci Flow, Self-Similar Solution and Futaki Invariant International conference

Ryoichi Kobayashi

Mini-course and Workshop - Topology of Algebraic Varieties

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Event date： 2012.9

Language：English   Presentation type：Oral presentation (general)

Country：Canada

We introduce Hamiltonian-K¥"ahler Ricci flow. This is obtained by modifying K¥"ahler-Ricci flow so that it preserves any K¥"ahler class.
We introduce its self-similar solution and the Futaki-type obstruction for
its existence.

27. Hamiltonian Volume Minimizing Property of Maximal Torus Orbit in P^n(C) International conference

Ryoichi Kobayashi

8-th China-Japan Geometry Conference

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Event date： 2012.9

Language：English   Presentation type：Oral presentation (general)

Country：China

I proposed an idea of applying the theory of Legendrian distributions
to Hamiltonian minimality problem. I proved that any maximal torus
orbit in P^n(C) is volume minimizing under Hamiltonian deformation.

28. Global Hamiltonian Stability of $U(1)^n$ orbits in $¥Bbb P^n$ International conference

Ryoichi Kobayashi

Russian-German Conference on Several Complex Variables

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Event date： 2012.2 - 2012.3

Language：English   Presentation type：Oral presentation (invited, special)

Country：Russian Federation

We prove that any $¥text{¥rm U}(1)^n$-orbit in $¥Bbb P^n$ is global
Hamiltonian stable (Hamiltonian volume minimizing), i.e., volume
minimizing under Hamiltonian deformation. The idea is that :
(1) we extend one $¥text{¥rm U}(1)^n$-orbit to the moment torus
fibration $¥{T_t¥}_{t¥in¥Delta_n}$ and consider its Hamiltonian
deformation $¥{¥phi(T_t)¥}_{t¥in¥Delta_n}$ where $¥phi$ is a
Hamiltonian diffeomorphism of $¥Bbb P^n$ and then :
(2) we compare each $¥text{¥rm U}(1)^n$-orbit and its Hamiltonian
deformation by comparing the large $k$ asymptotic behavior of the
sequence of projective embeddings defined, for each $k$, by the basis
of $H^0(¥Bbb P^n,¥Cal O(k))$ obtained by semi-clasasical
approximation of the $¥Cal O(k)$-BS (Bohr-Sommerfeld) tori of the
Lagrangian torus fibrations $¥{T_t¥}_{t¥in¥Delta_n}$ and its
Hamiltonian deformation $¥{¥phi(T_t)¥}_{t¥in¥Delta_n}$.

29. Calabi-Yau singular perturbation and metrics of constant positive curvature on compact Riemann surfaces with conic singularities

Ruoichi Kobayashi

The 17-th International Symposium on Complex Geometry

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Event date： 2011.11

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

We propose an extension of the rersults of Bando, the author [BK2],
Tian, Yau [TY] and van Coevering [vC] on the existence of a complete
Ricci flat K¥"ahler metric on $X¥setminus D$ where $X$ is a Fano
manifold and $D$ is a smooth hypersurface satisfying
$c_1(X)=¥alpha [D]$ with $¥alpha>1$ which admits a K¥"ahlerEinstein
metric (in [BK2] and [TY]) or whose link $S ¥subset N_{D/X}$ admits a
Sasaki-Einstein structure (in [vC]).
We remove the condition that $D$ admits a K¥"ahler-Einstein metric
or the link of $D$ admits a Sasaki-Einstein structure and prove that
there exists a complete asymptotically conical Ricci-flat K¥"ahler
metric on $X¥setminus D$. Our approach is based on a certain singular
perturbation of the Calabi-Yau theorem [Y1] on the existence of a K¥"ahler metric with a prescribed Ricci form (or the volume form).

30. Calabi-Yau singular perturbation and its geometric application

Tambara Complex Analysis Workshop

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Event date： 2011.9

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

31. Asymptotically conical K\"ahler metrics with prscribed Ricci form and its application to the fundamental group of varieties of general type International conference

USTC Geometry Seminar

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Event date： 2010.11

Language：English   Presentation type：Oral presentation (invited, special)

I introduced an exiarence theorem for an asymptotically
conical K\"ahler metric on affine algebraic manifold
and apply the it to establishing a sufficient condition
for the fundamental group of a variety of general type
to be finite.

32. Asymptotically conical Kaehler metrics with prescribed Ricci form and applications

16-th International Symposium on Complex Geometry

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Event date： 2010.10

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

I introduced an exiatence theorem (with estimates)
for asymptotically conical Kaehler metrics with
prescribed Ricci tensor on affine algebraic manifolds.
Then I apply the existence theorem to establish a
sufficient condition for the finiteness of the fundamental
group for projective varieties of general type.

33. Nevanlinna-Galois Theory for Algebraic Minimal Surfaces

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Event date： 2010.10

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

34. Asymptotically conical Ricci-flat K\"ahler metrics on affine algebraic manifolds International conference

6-th Geometry Conference for friendship between China and Japan

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Event date： 2010.9

Language：English   Presentation type：Oral presentation (invited, special)

We propose an extension of my previous rersult with Bando
(1989) and Tian-Yau (1989) and van Coevering (2009) on the
existence of a complete Ricci flat K\"ahler metric
on $X\backslash D$ where $X$ is a Fano manifold and $D$ is
a smooth hypersurface satisfying $c_1(X)=\alpha [D]$ with
$\alpha>1$ which admits a K\"ahler-Einstein metric (in
Bando-K and Tian-Yau) or whose link $S \subset N_{D/X}$
admits a Sasaki-Einstein structure (in [vC]). We remove the
condition that $D$ admits a

35. Asymptotically conical Ricci-flat K\"ahler metrics on affine algebraic manifolds International conference

5-th Pacific Rim Complex & Symplectic Geometry

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Event date： 2010.7

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

I proposed a general existence result of asymptotically
conical Ricci-flat K\"ahler metrics on certain affine
algebraic manifolds. This generalizes results of
Bando-Kobayashi and van Coevering.

36. Localization via iteration of parabolic translations and period condition for algebraic minimal surfaces

Geometry and Something

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Event date： 2009.11

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

37. Localization via parabolic translations, LLD and algebraic minimal surfaces

Holomorphic mappings and related Diophantine approximation

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Event date： 2009.10

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

38. Nevanlinna-Galois theory for algebraic minimal surfaces International conference

XXIst Rolf Nevanlinna Colloquium

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Event date： 2009.9

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

39. A construction og a transversal Ricci flow unstable cell centered at a a transversal Einstein metric on a certain fiber bundle over the twistor space of positive quaternion K\"ahler manifolds

Geometry and Something 2008

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Event date： 2008.11

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

40. Ricci flow unstable cell arising from the collapse of the twistor space of positive quaternion Kaehler manifolds International conference

Extremal Kaehler metrics and Kaehler Ricci flow

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Event date： 2008.3

Language：English   Presentation type：Oral presentation (invited, special)

I compared the canonical deformation metrics and the Chow-Yang metrics defined on the twistor space of positive quaternion Kaehler manifolds. I showed that the Ricci flow unstable cell
centered at a KE metric arises only in the family of Chow-Yang metrics.

41. Ricci flow unstable cell arising from the collapse of the twistor space of positive quaternion Kaehler manifolds

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Event date： 2008.2

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

42. Ricci flow unstable cell on the twistor space of positive quaternion Kaehler manifolds

Complex Geometry in Osaka

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Event date： 2007.11

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

43. Ricci flow unstable cell centered at a Kaehler-Einstein metric on the twistor space of positive quaternion Kaehler manifolds

The 13-th International Symposium on Complex Geometry, Sugadaira

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Event date： 2007.10

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

Perelman's local non collapsing theorem yields an ancient
solution of the Ricci flow by approproately rescaling the
Ricci flow space time when a singularity develops aty finite
time. This encodes all information of the singularity. Since
a parabolic equation is hardly extended to minus infinite
time, this situation is useful in various classification
problem. In this lecture I showed what happens if we apply
this idea to the twistor space of positive quaternion
Kaehler manifolds.

44. Some dynamical property of the Ricci flow on the twistor space of positive quaternion Kaehler manifolds International conference

Geometric quantization and related geometry (at Steklov Mathematical Institute, moscow)

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Event date： 2007.9

Language：English   Presentation type：Oral presentation (invited, special)

Perelman proposed an idea of viewing Ricci flow as a
gradient flow and proved that the Kaehler Ricci flow
solutions on a KE manifold form a stable cell.
I constructed an unstable cell around the KE metric on the
twistor space of positive QK manifold which consists of 2
parameter family of non Kaehler Ricci flow solutions. The existence of this unstable cell yields a limit formula
which stringly supports the LeBrun-Salamon conjecture on
the uniformization of positive QK manifold.

45. Ricci flow on the twistor space of positive quaternion Kaehler manifolds and its application International conference

Pacific Rim Complex Geometry Conference 2007

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Event date： 2007.8

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

I showed the construction of a 2-parameter family of
Riemannian metrics containing the Kaehler-Einstein metrics
which is foliated by the trajectories of the (non-Kaehler)
Ricci flow solutions. I showed a supporting limit formula
supporting the LuBrun-Salamon conjecture on the uniformization of positive quaternion Kaehler manifolds.

46. Nevanlinna-Galois Theory for pseudo-algebraic minimal surfaces International conference

Holomorphic mappings, Kobayashi hyperbolicity and Diophantine approximation (U. Tokyo)

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Event date： 2007.7

Language：English   Presentation type：Oral presentation (invited, special)

Country：Japan

### KAKENHI (Grants-in-Aid for Scientific Research) 7

1. Sacal-flat complete Kaehler metrics and K-stability at infinity

2014.4 - 2016.3

Grant-in-Aid for Scientific Research

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Authorship：Principal investigator

2. リッチ形式の局所化と漸近的チャウ安定ファノ多様体における反標準因子の存在について

2011.4 - 2014.3

科学研究費補助金

小林亮一

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ファノ多様体上のケーラー計量の1パラメータ族のリッチ曲率
の局在過程そのものを点とみなすようなモジュライ空間を構
成し，その上のベクトル束の非自明性を，リッチ曲率の局在
過程の解析的性質に読み替える枠組みを与える「幾何解析」
を構築して，漸近的チャウ安定ファノ多様体における次数の
低い因子の存在を示すことが，本研究の新しい着想である．

3. 対称空間の双対性に基づくケーラー・アインシュタイン計量の構成とその漸近解析

2008

科学研究費補助金  萌芽研究，課題番号：20654008

小林　亮一

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Authorship：Principal investigator

4. Statistical Laws in Geometry

2005.4 - 2008.3

Grant-in-Aid for Scientific Research

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Authorship：Principal investigator

5. Riemannian analogue of thermostat and Harnack Inequality

2005.4 - 2007.3

Grant-in-Aid for Scientific Research

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Authorship：Principal investigator

6. Dynamical approach to extremal Kaehler metrics

2002.4 - 2004.3

Grant-in-Aid for Scientific Research

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Authorship：Principal investigator

7. Toward discretazation of Nevanlinna Theory

2001.4 - 2004.3

Grant-in-Aid for Scientific Research

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Authorship：Principal investigator

### Teaching Experience (On-campus) 2

1. Linear Algebra I

2011

2. Linear Algebra II

2011

### Teaching Experience (Off-campus) 7

1. 代数的極小曲面の Osserman 理論とその量子化

2016.4 - 2017.3 Kyushu University）

2. 幾何学特論

2015.4 - 2016.3 Tohoku University）

3. 微分幾何学特論第一

2011.4 - 2012.3 Tokyo Institute of Technology）

4. リッチフローの３次元カッパ解の空間のコンパクト性

2009.4 - 2010.3 Waseda University）

5. ３次元リッチフローの標準近傍定理

2009.4 - 2010.3 The University of Tokyo）

6. リッチフローの非局所崩壊定理

2007.4 - 2008.3 Kyushu University）

7. 2015.4 - 2016.3 Meijo University）