International Workshop on Recent Development in Geometry

Date 16(Thu)-17(Fri) October, 2008
Place Mathematical Institute, Tohoku University (東北大学理学部数学):Kawai Hall (川井ホール)


Zhu Xiping (Zhongshan U. 中山大学, China)
"The Ricci Flow with Surgery in Four-manifolds"
In the middle of 1990's, Hamilton developed a theory of the Ricci flow with surgery and claimed a classification to compact four-manifolds of positive isotropic curvature with no incompressible space forms. In 2006, based on the works of Perelman, Chen and the speaker obtained a complete proof of the Hamilton's classification. Recently we can further give a complete classification to all compact four-manifold with positive isotropic curvature. The talk will describe this recent work.
Hu Jianxun (Zhongshan U. 中山大学, China)
"Gromov-Witten invariant and birational geometry"
In this talk, I will first introduce Gromov-Witten invariant and its degenertion formula.Then I want to talk about the birational equivalence in symplectic geometry and characterization of uniruled symplectic manifold. Finally I will talk about the birational cobordism invariance of uniruled symplectic manifolds. This is a joint work with Tian-Jun Li and Yongbin Ruan.
B. Palmer (Idaho Univ., USA)
"Anisotropic Surface Energies"
Anisotropic surface energies assign a value to a surface which depends on the direction of the surface at each point. We will discuss several types of anisotropic surface energies and their applications. These will include recent results about anisotropic capillary problems.
M. Ishida (石田政司)(Sophia U. 上智大)
"The normalized Ricci flow on four-manifolds and exotic smooth structures"
In this talk, from the gauge theoretical point of view, we will discuss the existence or non-existence of long time solutions with uniformly bounded curvature of the normalized Ricci flow in dimension 4. In particular, we would like to point out that the difference between existence and non-existence of such solutions of the normalized Ricci flow strictly depend on the choice of smooth structure.
R Kobayahi (小林亮一) (Nagoya U. 名古屋大)
"A family of new metrics on the twistor space of quaternion K"ahler manifold and their behavior under the Ricci flow"
I will construct a family of new metrics on the twistor space of quaternion K"ahler manifold and study their behavior under the Ricci flow. Comparison with the canonical deformation metrics on the twistor space and with the standard product metrics on the product of spheres will be presented. Abstract[PDF]
S. Sakaguchi (坂口 茂)(HIroshima U. 広島大)
"Stationary isothermic surfaces and a characterization of the hyperplane"
Consider an entire graph $S$ of a continuous real function over $ \mathbb R^{N-1}$ with $N \ge 3$. Let $\Omega$ be a domain with boundary $S$ in $\mathbb R^N$. Consider the heat flow with initial temperature $0$ and boundary temperature $1$ in $\Omega$. The problem we consider is to characterize $S$ such that there exists a stationary isothermic surface in $\Omega$. We show that $S$ must be a hyperplane under some general conditions on $S$. This is related to Liouville or Bernstein type theorems for some elliptic Monge-Amp \`ere type equation. Abstract[PDF]
M. Toda (戸田正人) (Ochanomizu U. お茶の水大)
"Scaling limits of the Ricci flow and monotone quantities"
I shall explain that the entropy bounds or the kappa-noncollapsing property is not enough to carry out the satisfactory scaling analysis of the Ricci flow of dimension four and higher. I shall also disscus some attempts to find a new monotone quantity which meets this purpose.
T. Yokota (横田 巧)(Tsukuba U. 筑波大)
"Perelman's reduced volume and gap theorem for the Ricci flow"
In this talk, we consider the asymptotic limit of Perelman's reduced volume for ancient solutions to the Ricci flow. We will show that if the asymptotic reduced volume is sufficiently close to that of the Gaussian soluton, then the ancient solution is isometric to the Euclidean space for all time. As an application, the case of the gradient shrinking Ricci solitons is also discussed.

K. Kenmotsu (剱持勝衛) Tohoku U.
R. Kobayashi (小林亮一) Nagoya U.
R. Miyaoka (宮岡礼子) Tohoku U.

Contact :
r-miyaok (at)

This Workshop is supported by Grants-in-Aid for Scientific Research 19204006 (R. Miyaoka).
製作 宮岡 Last updated on 11/Aug./2008