The 5th Geometry Conference for Friendship of Japan and China
Abstract

Orbifold Relative GWInvariants and Applications

Bohui Chen (Sichuan University)

(Abstract)
In this talk,
we generalize the relative GromovWitten invariant theory
to orbifolds and give some applications.

Gevery Regularity of Subelliptic MongeAmpere Equations in the Plane

Hua Chen (Wuhan University)

(Abstract)
In this talk, we shall consider the regularity problem
for a class of twodimensional degenerate MongeAmpere equations.
We know that, in this case, the solution is smooth
if the principal curvature of the solution is strictly positive
and the order of degenerate point for the coefficient of the equation is finite.
Here we prove that,
under the same conditions, the solution would be Gevery regularity.

The Ricci Curvature In Finsler Geometry

Xinyue Cheng (Chongqing University of Technology)

(Abstract)
The Ricci curvature plays a very important role in Finsler geometry.
In this talk, we introduce the Ricci curvature and its some applications
in Finsler geometry,
including the role of the Ricci curvature in Finsler projective geometry,
the volume comparison in Finsler geometry and Ricci curvature of Randers metrics.

Using topology to explore DNA packing

Yuanan Diao (University of North Carolina at Charlotte)

(Abstract)
In this talk, we will discuss how topology
(more specifically knot theory)
is used to explore and to provide insights
about basic properties of DNA and of its packing principles.
We will also discuss various models used to describe the DNA packing
geometry in many bacteriophages and some animal.

Prescribed curvature representation and its applications

Qing Ding (Fudan University)

(Abstract)
In this talk, we introduce the geometric concept of PDEs with prescribed curvature
representation in the category of YangMills theory,
which generalizes the wellknown concept of integrable equations
with zero curvature representation.
As the applications,
we use this new geometric tool to study the dynamical properties
of the nonintegrable discrete (semicalssical) Heisenberg model
${S_n}_t={S_n} \times (S_{n+1}S_{n1})$
and the (classical) Schrodinger maps from the Euclidean plane
to the hyperbolic 2space $H^2$.

A PoincareHopf index formula for complex vector fields

Huitao Feng (Nankai University)

(Abstract)
For two complex vector bundles admitting a homomorphism
with isolated singularities between them,
we establish a PoincareHopf type formula
for the difference of the Chern character numbers of these two vector bundles.
As a consequence, we extend the original PoincareHopf index formula
to the case of the complex vector fields.

The volume growth of hyperkaehler manifolds of type $A_{\infty}$

Kota Hattori (University of Tokyo)

(Abstract)
Hyperkaehler manifolds of type $A_{\infty}$ were first constructed by
GibbonsHawking ansatz due to Anderson, Kronheimer and LeBrun.
These manifolds are noncompact and their homology groups are infinitely generated.
We focus on the volume growth of these hyperkaehler metrics.
Here, the volume growth is asymptotic behavior of the volume
of a ball of radius r>0 where r goes to infinity with the center fixed.
There are known examples of hyperkaehler manifolds
whose volume growth is r^4 (ALE space) or r^3 (TaubNUT space).
In this talk we show that there exists a hyperkaehler
manifold of type $A_{\infty}$
whose volume growth is r^c for a given 3 < c < 4.

Ricci flow, simplicial volume, and smooth structures

Masashi Ishida (Sophia University)

(Abstract)
It is conjectured by Fuquan Fang, Yuguang Zhang and Zhenlei Zhang
that the existence of nonsingular solutions
to the normalized Ricci flow on smooth closed 4manifolds
with nontrivial Gromov's simplicial volume and negative Perelman's invariant
implies the GromovHitchinThorpe type inequality.
This conjecture is still open.
In this talk, we shall discuss the existence of
closed topological 4manifolds with nontrivial Gromov's simplicial volume
and satisfying the GromovHitchinThorpe type inequality,
but admitting infinitely many exotic smooth structures
for which Perelman's invariant is negative
and there is no nonsingular solution to the normalized Ricci flow
for any initial metric.
In particular, the main result of this talk tells us
that the converse of the conjecture dose not hold in general.
We use the SeibergWitten monopole equations to prove the main result.

Modeling of morphogenesis in development

Yoh Iwasa (Kyushu University)

(Abstract)
We start our life from a single cell, named fertilized egg, and
develop to a adult body with many complex structures. I will speak on
recent attempts of modeling morphogenesis in development. In
particular, I will speak two examples of modeling of organ growth:
limb (wing) bud formation of chick and branching pattern formation of
kidney ducts. Finally I will speak on the analysis of traveling wave
of gene expression in zebrafish somitogenesis.

The symplectic volume of the moduli space of spatial polygons

Yasuhiko Kamiyama (Ryukyu University)

(Abstract)
The moduli space of spatial polygons is a typical example
in symplectic geometry and toric topology.
The symplectic volume of the space was studied
by many mathematicians using various approaches.
I will survey these researches.

A rough equivalence among partial differential equations

Tsuyoshi Kato (Kyoto University)

(Abstract)

Real hypersurfaces in a complex space form and the generalized TanakaWebster connection

Mayuko Kon (Hokkaido University)

(Abstract)
In this talk I will present the results for the curvature tensor and the
Ricci tensor with respect to the generalized TanakaWebster connection of
a real hypersurface in a complex space form.
The generalized TanakaWebster connection
for a real hypersurfaces of Kaehlerian manifolds
was studied by J. T. Cho.
It coincids with the TanakaWebster connection
if the associated CRstructure of the real hypersurface is
pseudoHermitian and strongly pseudoconvex.

Random groups and scaling limit argument

Takefumi Kondo (Kobe University)

(Abstract)

A KazdanWarner type identity and a class of curvature invariants $v^{2k}$

Haizhong Li (Tsinghua University)

(Abstract)
In this talk, we present a KazdanWarner type identity involving the $v^{2k}$
curvature and a Killing vector field on a compact manifold.
In the special case when the Riemannian manifold is locally conformally flat,
the result reduces to the wellknown result.

Non existence of quasiharmonic spheres

Jiayu Li (Chinese Academy of Sciences)

(Abstract)
Let $M$ and $N$ be compact Riemannian manifolds.
To prove the global existence and convergence of the heat flow
for harmonic maps between $M$ and $N$,
it suffices to show the nonexistence of harmonic spheres
and nonexistence of quasiharmonic spheres.
In this talk, we prove that,
if the universal covering of $N$ admits a nonnegative
strictly convex function with polynomial growth,
then there are no quasiharmonic spheres nor harmonic spheres.
This generalizes the famous EellsSampson's theorem.

On Lagrangian submanifolds in complex hyperquadrics

Hui Ma (Tsinghua University)

(Abstract)
The Gauss map of any oriented isoparametric hypersurface of the sphere
defines a minimal Lagrangian submanifold in the complex hyperquadric.
In this talk, we determine the Hamiltonian stability of
ALL compact minimal Lagrangian submanifolds embedded
in complex hyperquadrics
obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres.
The relation between the Gauss image construction and the conormal
bundle construction for Lagrangian submanifolds in complex
hyperquadrics will also be discussed.
This talk is mainly based on the joint work with Professor Yoshihiro Ohnita.

An approach to DonaldsonTianYau's Conjecture

Toshiki Mabuchi (Osaka University)

(Abstract)
DonaldsonTianYau's Conjecture asks whether
a polarized algebraic manifold $(M,L)$ admits a CSC K\"ahler metric
if and only if $(M,L)$ is Kstable.
In this talk,
we shall give our recent approach to the existence
problem for CSC K\"ahler metric.

The moduli space of transverse CalabiYau structures on foliated manifolds

Takayuki Moriyama (Kyoto University)

(Abstract)
We develop a moduli theory of transverse structures
given by closed forms on foliated manifolds.
We show that the moduli space of transverse CalabiYau structures
is a Hausdorff and smooth manifold if the foliation is taut.
In this talk, we will give some examples of
transverse CalabiYau structures on foliated manifolds.

Wave equations and the LeBrunMason twistor correspondence

Fuminori Nakata (Tokyo Institute of Technology)

(Abstract)

Invariant geometric flows and integrable systems

Changzheng Qu (Northwest University)

(Abstract)
In this talk, we shall discuss the relationship
between invariant geometric flows and integrable systems.
It is shown that many integrable systems
are associated with the invariant geometric flows in some geometries.
The geometric interpretation to properties of integrable systems
such as the Backlund transformation,
Miura transformation and biHamiltonian structure etc
are given in terms of the geometric flows.

Finslerian models in biology

Sorin Sabau (Tokai University)

(Abstract)
I will prsent two topics in this talk.
Firstly, some Finslerian type models introduced in ecology,
and secondly, the socalled KosambiCartanChern theory.
The second topic is a Finslerian alternative to the classical notion
of Lyapunov stability of dynamical systems
and it is used in various fields of the science nowadays.

Prolongations of canonical systems on Jet spaces

Kazuhiro Shibuya (Hiroshima University)

(Abstract)
We will introduce the prolongatoin procedure
for differential systems given by E.Cartan to study PDEs.
In this tahk, we will consider a generalization of Monster Goursat manifolds
which are obtained by prolongations of canonical systems on Jet spaces.
It will be important to study geometric solutions of PDEs.

Tangle analysis of sitespecific recombination

Koya Shimokawa (Saitama University)

(Abstract)
Action of sitespecific recombinase can be analysed using
tangle model introduced by Ernst and Sumners.
In this talk we apply this method and give topological characterizations
of actions of several sitespecific recombinases.

A splitting theorem for weighted Alexandrov spaces

Takashi Shioya (Tohoku University)

(Abstract)

Selection for Complexity can Induce Modularity

Robert Sinclair (Okinawa Institute of Science and Technology)

(Abstract)
The animals, plants, birds and fish
we know seem to belong to a progression towards ever greater complexity.
We can identify the parts of the more complex organisms,
and know that these parts are sometimes exchangeable
by grafting or transplantation.
This suggests modular design.
Are the higher organisms more modular than, say,
mushrooms due to more complex challenges they must face?
For an abstract model,
a synchronous boolean network model with only inhibitory interactions,
we provide a computer proof to demonstrate
that the greatest complexity for a given system size is only achievable
if the maximal network is disjoint,
for systems not less than a critical size.
This is a mathematical approach to a biological problem
which required the development of dedicated software.
A number of purely mathematical conjectures arise from this work.

Tissue invasion: early stage of tumor metastasis and its mathematics

Takashi Suzuki (Osaka University)

(Abstract)
Invasion arises in the early stage of tumor metastatis
and its control has been the important issue in medical science.
This process is composed of three factors
 actin reorganization, ECM degradation, and adhesive regulation.
We shall describe two mathematical approaches to this object,
inverse source identification and mathematical modeling.
In the former case we adopt several new mathematical tools,
homology and parallel optimization to take advances in cancer diagnosis.
In the latter we show the notion of a bidirected modeling
whereby new mean field equation is formulated.
Numerical simulations and mathematical analysis of these models are also presented.

Pattern formation on surfaces  from a biological point of view

Izumi Takagi (Tohoku University)

(Abstract)

Wasserstein geometry of nonlinear FokkerPlanck equations

Asuka Takatsu (Tohoku University)

(Abstract)
In this talk, inequalities associated with the nonlinear FokkerPlanck equations
will be introduced in order to describe relations among
functionals defined on pairs of probability measures,
called generalized relative entropy, Fisher information, and Wasserstein distance.
I will discuss the importance of the $q$Gaussian measures in these inequalities.

Isoparametric hypersurface in a Reimannian manifold

Zizhou Tang (Beijing Normal University)

(Abstract)
Isoparametric hypersurfaces in various Riemannian manifolds will be studied,
especially in Milnor exotic spheres.

KahlerEinstein metrics and compactness theorem on Fano manifolds

Gang Tian (Peking University and Princeton University)

(Abstract)

Geometry of wave fronts

Masaaki Umehara (Osaka University)

(Abstract)

Moebius CMCsurfaces in S^3

Changping Wang (Peking University)

(Abstract)
Let x: M \to R^3 be a surface without umbilical point.
Let \xi be the sphere tangent to x with radius 1/H,
where H is the mean curvature of x.
Then \xi: M \to S^4_1 is a surface in the deSitter space,
called the mean curvature sphere of x.
Moreover, the mean curvature vector of \xi is lightlike.
Conversely, any surface \xi: M \to S^4_1 with lightlike mean curvature vector
is the mean curvature sphere of a surface x: M \to R^3.
A surface x: M \to R^3 is called Moebius CMC surface
if its mean curvature sphere
\xi: M \to S^4_1 is a surface with parallel mean curvature vector.
It is a generalization of Willmore surfaces.
In this talk, we classify
all Moebius CMCsurfaces with constant Moebius curvature:
they are Moebius equivalent to either minimal surfaces in R^3,
or CMC1 surfaces in H^3, or a cirular cylinder,
a circular cone, a revolution torus in R^3.
We classify also all surfaces x: M \to R^3
such that its mean curvature sphere \xi
intersects a fixed plane with constant angle.

The regularity of harmonic maps into spehres and applications to Bernstein problems

Yuanlong Xin (Fudan University)

(Abstract)
We show the regularity of, and derive apriori estimates for
(weakly) harmonic maps from a Riemannian manifold
into a Euclidean sphere under the assumption
that the image avoids some neighborhood of a halfequator.
The proofs combine constructions of strictly convex functions
and the regularity theory of quasilinear elliptic systems.
We apply these results to the spherical and Euclidean Bernstein problems
for minimal hypersurfaces,
obtaining new conditions under which compact minimal hypersurfaces
in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

Harmonic metrics on unipotent bundles over quasicompact Kaehler manifolds

Yihu Yang (Tongji University)

(Abstract)
In this talk, I will show the existence of harmonic metrics
on unipotent bundles over quasicompact Kaehler manifolds.

Towards a new sphere theorem via the Ricci flow

Takumi Yokota (University of Tsukuba)

(Abstract)
The Ricci flow is one of the powerful tools in Riemannain geometry.
We can deform a given positively curved Riemannain manifold
to a rounder one by solving the Ricci flow equation,
which is a heattype equation for evolving Riemannian metrics.
In this talk, I will talk about an approach
to apply the Ricci flow to prove a new sphere theorem.

Geometric quantization on manifolds with boundary

Weiping Zhang (Nankai University)

(Abstract)
We describe our recent joint work with Xiaonan Ma
where we resolve a conjecture of M. Vergne
on the geometric quantization formula on noncompact manifolds
with proper moment maps.

Classification of Fourdimensional Manifolds with Positive Isotropic Curvature

Xiping Zhu (Sun Yatsen University)

(Abstract)
Around 1980, Schoen and Yau used minimal surface theory to
obtain their famous classification to threedimensional manifolds
with positive scalar curvature.
Nevertheless, up to now,
the understanding to positively curved fourdimensional manifolds is still poor.
For example, one does not know if $S^2 \times S^2$ admits
a metric of positive sectional curvature (Hopf conjecture);
one does not have the classification of fourdimensional Einstein manifolds
with positive scalar curvature.
Recently, inspired by the HamiltonPerelman theory,
Chen, Tang and the speaker used the Ricci flow with surgery
to obtain a complete classification to fourdimensional manifolds
with positive isotropic curvature.
We proved that a compact fourdimensional manifold
with a metric of positive isotropic curvature
if and only if it is diffeomorphic to the standard $S^4$,
or the standard $RP^4$,
or a cocompact metric quotient of the standard $R \times S^3$,
or a connected sum of a finite number of them.
Note that it was earlier proved by Hamilton that
any exotic $RP^4$ does not admits a metric of positive isotropic curvature.
As a byproduct, we also obtain a complete classification to
compact, locally conformally flat fourdimensional manifolds
with positive Yamabe invariant.
In this talk I will discuss these classification theorems.