The program

CLICK HERE FOR THE SCHEDULE OF LECTURES

The schedule of lectures is now in final form. Only minor modifications are anticipated; speakers who are affected by such changes will be contacted individually. The program distributed at registration will supersede this version.

It is the practice of the Mathematical Society of Japan to publish the proceedings of each MSJ-IRI in the series ADVANCED STUDIES IN PURE MATHEMATICS, in cooperation with the American Mathematical Society. The organisers of the 9th MSJ-IRI intend to follow this custom. According to the MSJ rules, the proceedings volume should be an "up to date guide of lasting interest...containing good surveys of the area covered, not being merely a collection of individual articles" and all articles and surveys are to be refereed.

A separate volume containing more specialized articles and research announcements may also be published (in a format yet to be decided).

Participants are invited to submit contributions to these proceedings volume(s). For the ASPM volume, priority will be given to articles which focus on the interactions between integrable systems and differential geometry.

More detailed information about the proceedings will be sent to all speakers after the conference.

Speakers, titles and abstracts

The index of an isolated umbilical point on a real-analytic surface

In this talk, we will describe one way of computing the index of an isolated umbilical point on a real-analytic surface of a certain type, which contains all the real-analytic, special Weingarten surfaces.

Ricci curvature and Staeckel systems

In 1978 Alfred Gray suggested several generalizations of Einstein manifolds. His idea was as follows. An Einstein manifold has parallel Ricci tensor, and conversely, a Riemannian manifold with parallel Ricci tensor is locally a Riemannian product of Einstein manifolds. Thus the Einstein condition is essentially equivalent to the parallelity of the Ricci tensor. Now consider the covariant derivative $\nabla S$ of the Ricci tensor $S$ of a Riemannian manifold $M$ as a tensor field of type $(0,3)$. The algebraic curvature identities and the second Bianchi identity of the Riemannian curvature tensor of $M$ imply certain algebraic equations for $\nabla S$. Let $V$ be the real vector bundle over $M$ whose fibre $V_p$ at $p \in M$ consists of all trilinear maps on $T_pM$ satisfying these algebraic equations. The orthogonal group $O(n)$, $n = \dim M$, acts in a standard way on $V$, turning each fibre $V_p$ into an $O(n)$-module. If $n\geq 3$ then $V_p$ decomposes into three irreducible $O(n)$-modules, inducing a direct sum decomposition $V = V_1 \oplus V_2 \oplus V_3$. This decomposition provides naturally several types of generalizations of the Einstein condition. The bundle $V_1$ characterizes Riemannian manifolds with harmonic curvature, and $V_2$ characterizes Riemannian manifolds for which the Ricci tensor is a Killing tensor. The covariant derivative $\nabla S$ of the Ricci tensor $S$ of a Riemannian manifold $M$ is a section in $V_3$ if and only if $$ \nabla \left( S - {1 \over 2n-2}sg \right) = {n-2 \over 2(n+2)(n-1)} ds \odot g\ , $$ where $s$ is the scalar curvature, $g$ the Riemannian metric and $n$ the dimension of $M$. Many examples are known for the first two types of manifolds, but the PDE system characterizing the third bundle appears to be rather awkward and only a little is known about it. The purpose of the talk is to present a local classification of all 3-dimensional Riemannian metrics solving this PDE system. The method is to relate this problem to the classical theory about complete integrability of the geodesic Hamilton-Jacobi equation by separation of variables.

Circle patterns and integrable systems

Circle patterns approximating holomorphic mappings are studied. It is shown that they correspond to discrete integrable systems. Methods from the theory of integrable systems are applied to study local and global properties of these circle patterns.

Pluecker formulae and the Toda equations

The full flag manifold $$SU(n+1)/T^n=\{(L_0,\ldots , L_n)\ :\ {\bf C}^{n+1}=L_0\perp \ldots \perp L_n\}$$ is an $(n+1)$-symmetric space, that is to say at each point $x$ there is a symmetry $\tau_x$ of $SU(n+1)/T^n$ of order $n+1$ which has $x$ as an isolated fixed point. A map $\phi$ from a Riemann surface $S$ to $SU(n+1)/T^n$ is {\it $\tau$-adapted} if $\phi$ is a branched conformal immersion and, for each $z\in S$, the differential of $\tau_{\phi(z)}$ maps $d\phi(T_zS)$ into itself by rotation through $2\pi/(n+1)$. Such maps correspond locally to solutions of the 2-dimensional Toda equations.

One way in which such maps arise is via a linearly full holomorphic curve $\psi(z)=[f(z)]$ in ${\bf C}P^n$. For, if $F_0,\ldots ,F_n$ is the result of applying Gram-Schmidt orthonormalisation to the sequence $f,f^{(1)},\ldots f^{(n)}$ of derivatives of $f$ then $[F_0,\ldots,F_n]$ is $\tau$-adapted.

When $S$ is compact, the classical Pl\"ucker formulae connect the Euler characteristic of $S$ and two sets of integers which arise from the geometry of the osculating curves of $\psi$. We will discuss a generalisation of these formulae to $\tau$-adapted maps into $SU(n+1)/T^n$ and, more generally, into $G/T$, where $T$ is the maximal torus of a compact simple Lie group.

Isothermic submanifolds in symmetric R-spaces

The classical theory of isothermic surfaces studied by Christoffel \cite{Chr67}, Darboux \cite{Dar99,Dar99A} and Bianchi \cite{Bia05,Bia05A} has enjoyed a renaissance of interest in recent years as a result of the observation of Cie\'sli\'nski--Goldstein--Sym \cite{CieGolSym95} that these surfaces constitute an integrable system.

A principal feature of the theory is that isothermic surfaces (like Willmore surfaces \cite{Tho23,Bry84}) are conformally invariant while the integrability manifests itself in a rich transformation theory with B\"acklund-type transforms, the conformal Ribeaucour transforms of Darboux \cite{Dar99}, and a spectral deformation, the $T$-transform of Bianchi \cite{Bia05} and Calapso \cite{Cal03}. Moreover, much of this theory can be explained in terms of the \emph{curved flat} integrable system of Ferus--Pedit \cite{FerPed96}.

I explained in \cite{Bur00} how this whole story goes through for isothermic surfaces in $\R^n$ (see, also, Schief \cite{Schpe} for independent results in this direction).

In this talk, I shall describe joint work with Franz Pedit and Ulrich Pinkall which offers another generalisation of the theory: in view of the conformal invariance of isothermic surfaces, the natural setting for the theory is that of submanifold geometry in the conformal $n$-sphere but it turns out that the relevant geometry is that of the sphere \emph{qua} symmetric $R$-space.

Symmetric $R$-spaces are real analogues of Hermitian symmetric spaces that can be characterised according to Nagano \cite{Nag65} as compact Riemannian symmetric spaces that admit a Lie group of diffeomorphisms strictly larger than the isometry group. Examples include: \begin{itemize} \item the sphere with its group of conformal diffeomorphisms; \item real projective spaces and Grassmannians under the projective action of the special linear group; \item Hermitian symmetric spaces with their groups of biholomorphisms. \end{itemize}

The algebraic structure of such spaces is exactly what is needed to define the notion of an \emph{isothermic submanifold} in such a way that the integrable structure of the classical theory carries through unchanged.

I shall sketch these ideas and describe an example (known since the 1930s!): an isothermic surface in the space of lines in projective $3$-space is the same as an $R$-congruence in the sense of Tzitz\'eica.

\bibitem{Bia05} L.~Bianchi, \emph{{R}icerche sulle superficie isoterme e sulla deformazione delle quadriche}, Ann.\ di Mat. \textbf{11} (1905), 93--157.

\bibitem{Bia05A} \bysame, \emph{{C}omplementi alle ricerche sulle superficie isoterme}, Ann.\ di Mat. \textbf{12} (1905), 19--54.

\bibitem{Bry84} R.L. Bryant, \emph{{A} duality theorem for {W}illmore surfaces}, J. Diff.\ Geom. \textbf{20} (1984), 23--53.

\bibitem{Bur00} F.E. Burstall, \emph{{I}sothermic surfaces: conformal geometry, {C}lifford algebras and integrable systems}, Preprint math.DG/0003096, 2000.

\bibitem{Cal03} P.~Calapso, \emph{{S}ulle superficie a linee di curvatura isoterme}, Rendiconti Circolo Matematico di Palermo \textbf{17} (1903), 275--286.

\bibitem{Chr67} E.~Christoffel, \emph{{U}eber einige allgemeine {E}igenshaften der {M}inimumsfl\"achen}, Crelle's J. \textbf{67} (1867), 218--228.

\bibitem{CieGolSym95} J.~Cie\'sli\'nski, P.~Goldstein, and A.~Sym, \emph{{I}sothermic surfaces in {${E}^3$} as soliton surfaces}, Phys.\ Lett.\ A \textbf{205} (1995), 37--43.

\bibitem{Dar99} G.~Darboux, \emph{{S}ur les surfaces isothermiques}, C.R. Acad.\ Sci.\ Paris \textbf{128} (1899), 1299--1305.

\bibitem{Dar99A} \bysame, \emph{{S}ur une classe de surfaces isothermiques li\'ees \`a la d\'eformations des surfaces du second degr\'e}, C.R. Acad.\ Sci.\ Paris \textbf{128} (1899), 1483--1487.

\bibitem{FerPed96} D.~Ferus and F.~Pedit, \emph{{C}urved flats in symmetric spaces}, Manuscripta Math. \textbf{91} (1996), 445--454.

\bibitem{Nag65} T.~Nagano, \emph{{T}ransformation groups on compact symmetric spaces}, Trans.\ Amer.\ Math.\ Soc. \textbf{118} (1965), 428--453.

\bibitem{Schpe} W.K. Schief, \emph{{I}sothermic surfaces in spaces of arbitrary dimension: {I}ntegrability, discretization and {B}\"acklund transformations. {A} discrete {C}alapso equation}, Stud.\ Appl.\ Math. (appear).

\bibitem{Tho23} G.~Thomsen, \emph{{U}eber konforme {G}eometrie {I}: {G}rundlagen der konformen {F}laechentheorie}, Hamb.\ Math.\ Abh. \textbf{3} (1923), 31--56.

Quantization of the Benny Hierarchy

We compute the free energy and G-function associated with the Benney hierarchy and then quantize the Benney hierarchy from genus zero to one-loop correction using Dubrovin-Zhang's topological field theory. On the other hand, we also quantize the Benney hierarchy from the Konopelchenco-Oevel bracket of the modified KP hierarchy. We find that after an appropriate differential substitution, they are matched up to O(\epsilon^4).

CMC surfaces

The goal of this talk is to discuss how loop groups can be used in differential geometry for the construction of classes of surfaces. In particular we will address some deformation techniques and how in this formalism examples of certain topological types can be incorporated.

Geometry of bihamiltonian evolutionary equations

An approach to Gromov-Witten invariants based on a classification problem in bihamiltonian geometry will be discussed.

Micallef [Mi] proved that full stable hyperelliptic minimal surfaces in Riemannian flat tori are holomorphic with respect to some orthogonal complex structures of tori. On the other hand, Oort and Steenbrink [O-S] proved that the Torelli mapping of the moduli of Riemann surfaces into the moduli of principally polarized abelian varieties is a locally closed immersion. I would like to talk about the theory of compact orientable branched minimal surfaces in flat tori and prove that these two statements are equivalent.

Theorem [Ej]. Micallef's result is equivalent to Oort-Steenbrink's result.

[Ej] N. Ejiri, A differential-geometric Schottky problem and minimal surfaces in tori, preprint.

[Mi] M. Micallef, Stable minimal surfaces in flat tori, Contemporary Math. 49(1986), 73-78.

[O-S] F. Oort and J. Steenbrink, The local Torelli problem for algebraic vurves, Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Springer, Berlin, 1979 (Reihard B\"olling).

Surfaces in 3-space possessing nontrivial deformations which preserve the Weingarten operator

The class of surfaces in 3-space which possess nontrivial deformations preserving principal directions and principal curvatures (or, equivalently, the Weingarten operator) was investigated by Finikov and Gambier as far back as 1933. In my talk I intend to review known results, demonstrate the integrability of the corresponding Gauss-Codazzi equations and draw parallels between this geometrical problem and the theory of compatible Poisson brackets of hydrodynamic type. It turns out that the coordinate hypersurfaces of n-orthogonal systems arising in the theory of compatible Poisson brackets of hydrodynamic type must necessarily possess deformations preserving the Weingarten operator.

Timelike surfaces with hamonic inverse mean curvature

As an analogue to harmonic inverse mean curvature (HIMC) surfaces in Euclidean space first introduced by A. Bobenko, we consider HIMC surfaces in Lorentzian space forms (joint work with J. Inoguchi).

Mirror symmetry and Floer homology

In this talk, I would like to survey a relation between Floer homology and Mirror Symmetry (homological Mirror Symmetry).

Given a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L$ in $M$ we construct a moduli space of Lagrangian submanifolds $\Cal M(L)$ (at least locally). For another Lagrangian submanifold $L'$, the family of Floer homologies $HF(L,L')$ is expected to give an object of the derived category of coherent sheaves on $\Cal M(L)$.

The product structures and boundary operators of Floer homology determine a function (distribution valued current with coefficient in some hom bundle) which satisfies a differential equation, the Maurer-Cartan equation:

\overline\partial\frak m_k + \sum \pm \frak m_{\ell} \circ \frak m_{k-\ell+1} = 0. \tag{1}$$

Deforming $M$ also gives a similar differential equation.

In other words, elements of Floer homology give solutions of (1).

I would like to explain some general ideas of how to construct such a system and to prove homological Mirror Symmetry, assuming convergence of various formal power series, and show how the quantum effect (which is caused by the presence of pseudo holomorphic disks) appears and what kinds of role it is supposed to play in the story.

In fact, it seems likely that one can prove a large part of the homological mirror symmetry conjecture (by Kontsevich) in the case when a special Lagrangian fibration exists, if we can show convergence of several formal power series.

Also I want to discuss several examples, such as a family of (special) Lagrangian tori in K3 surfaces, and genus two Riemann surfaces in $T^4$, which are special Lagrangian submanifolds. In the first example (tori in K3), we can observe some phenomenon to show how the complex structure of $\Cal M(L)$ is deformed by the quantum effect. In the second example (genus two Riemann surfaces in $T^4$), we can observe some phenomena where the holomorphic structure of the family of Floer homology is twisted by the quantum effect.

Rozansky-Witten invariants of log symplectic manifolds

Rozansky and Witten introduced a new invariant of knots and three manifolds by using hyperK\"ahler manifolds. This is a generalization of the Chern-Simons invariant, and hyperK\"ahler manifolds play the role of the Lie algebra in Chern-Simon theory, and the Bianchi identity plays the role of the Jacobi identity.

Kontsevich and Kapranov show that this invariant can be constructed from holomorphic symplectic structures of compact complex manifolds. Hyperk\"ahler manifolds can be considered as holomorphic symplectic manifolds for a complex structure. Rozansky and Witten also suggest that their invariant would be constructible for reduced SU$(2)$ monopole moduli space of charge $k$. However this monopole moduli space of charge $k$ is a noncompact hyperk\"ahler manifold and we can not apply Kapranov's approach to this case. In this talk we shall generalize this invariant to complex manifolds with logarithmic symplectic forms. A logarithmic symplectic form is by definition, a closed and nondegenerate logarithmic $2$ form. The reduced SU$(2)$ monopole moduli space of charge $k$ can be compactified as a logarithmic symplectic manifold. Our logarithmic approach is suitable for this monopole moduli space.

Soliton-like solutions of generalized self-dual Yang- Mills Flows in ${\bold R}^{1+2n}$

The self-dual Yang-Mills equations are generalized in ${\bold R}^{2n+1}(n\geq 2)$. Soliton-like solutions are obtained explicitly. Their asymptotic behaviours are elucidated.

From submanifold geometry to Kac-Moody algebras

We describe recent progress in the theory of isoparametric and equifocal submanifolds and discuss their relations to infinite dimensional geometry as well as to involutions of affine Kac-Moody algebras.

Hamiltonian stationary Lagrangian surfaces and integrable systems-1

We study Lagrangian surfaces in homogeneous four-dimensional symplectic Kaehler manifolds which are critical points of the area functional for Hamiltonian vector field deformations. This problem turns to be completely integrable. It leads to various descriptions with a dramatic simplification for surfaces in Euclidean space.

Discrete Hashimoto surfaces and a discrete smoke ring flow

It can be shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schr\"odinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with this Hashimoto or smoke ring flow. From the geometry of B\"acklund transformations for smooth and ``space discrete'' Hashimoto surfaces a discrete time evolution for closed polygons is derived that should be viewed as a doubly discrete Hashimoto flow. It is shown, that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion only under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.

Fano manifolds with contact structures

We study Fano manifolds with contact structures. A good example is the adjoint variety-the orbit of a highest weight vector in the projectivized adjoint representation of a simple algebraic group. It is conjectured that Fano manifolds with contact structures are the adjoint varieties. This is related to the problem in Riemannian geometry of whether quaternionic K\"aler manifolds are quaternionic symmetric spaces. We present several known partial results and prove that the conjecture holds under the assumption that the Fano manifold has a geometric structure modelled after the adjoint variety. We use the result of N. Tanaka and K. Yamaguchi on the differential systems and its special reductions, and the theory of rational curves in Fano manifolds.

The complete classification of pseudospherical congruences in ${\bold R}^{2+1}$

By using Darboux transformations all pseudospherical congruences in the Minkowski space ${\bold R}^{2+1}$ are determined. This also gives a method to construct all kinds of surfaces of constant curvature in ${\bold R}^{2+1}$.

Generalised Hitchin systems

In the study of algebraically integrable Hamiltonian systems, (roughly, systems of Abelian varieties), there is one class of system, which while not in any sense universal, seems to include as special cases most of the classical examples such as tops, finite gap cases of integrable p.d.e, etc. These systems are the generalised Hitchin systems. I will give a survey of the systems, definitions, basic properties and examples.

Recent results on semi-Riemannian submersions

We give classification theorems for semi-Riemannian submersions with totally geodesic fibres, analogous to Escobales's.

On compact non-Kaehlerian Hermite-Liouville surfaces

A compact Kaehler-Liouville surface is a compact Kaehler surface equipped with a Kaehler-Liouville structure, which is defined as a 2-dimensional vector space spanned by its energy function and a first integral of its geodesic flow satisfying certain conditions. A compact Kaehler-Liouville surface with the condition of properness has the property that its geodesic flow is completely integrable. The concept of the Kaehler-Liouville structure can be naturally generalized to the case of the hermitian surface; the Hermite-Liouville structure (the Hermite-Liouville surface) can be defined by the hermitian analogue of the definition of Kaehler-Liouville one. In this talk we will show that non-Kaehlerian Hermite-Liouville structures can be constructed on the complex projective plane with non-Kaehlerian metric by means of the complexification of the compact real Liouville surface. The compact Hermite-Liouville surface thus constructed also has the property of complete integrability of its geodesic flow. Besides, the non-Kaehlerian Hermite-Liouville structures can be constructed on the Hopf surface.

Integrable surfaces in Lorentzian geometry

A Lorentz surfaces is an orientable 2-manifold together with a Lorentzian conformal structure compatible with the orientation. Conformal immersions of a Lorentz surface in Lorentzian 3-manifolds are called "timelike surfaces".

Lorentzian conformal structures play important roles in the theory of integrable systems. For instance, the second fundamental form of a negatively curved surface in Euclidean 3-space determines a Lorentz conformal structure on the negatively curved surface. With respect to this conformal structure, the harmonicity of the Gauss map is equivalent to the constancy of the Gaussian curvature.

From the viewpoint of integrability, the study of timelike surfaces in Lorentzian space is an interesting subject. In fact, timelike constant mean curvature surfaces correspond to sinh-Gordon, Liouville or cosh-Gordon equations.

Timelike CMC surfaces

(isothermic) u_{xy}+\sinh u=0,
(null Hopf differential) u_{xy}+e^{u}/2=0,
(anti isothermic) u_{xy}+\cosh u=0.

This table may be considered as a "hyperbolic" analogue of constant mean curvature surfaces in hyperbolic 3-space:

CMC surfaces in H^3

(H^2>1) u_{z {\bar z}}+\sinh u=0,
(H^2=1) u_{z {\bar z}}+e^{u}/2=0,
(H^2<1) u_{z {\bar z}}++\cosh u=0.

In this talk, I would like to discuss Weierstrass type representations for timelike and spacelike CMC surfaces.

The lattice Toda field theory for simple Lie algebras

The discretization of the Toda field theory for any finite-dimensional simple Lie algebra is studied by introducing a Hamiltonian structure. We transform the Toda field equation into an equation for the $\tau$-function, and relate this $\tau$-funciton to the T-system (the functional relation between commuting families of the transfer matrices of solvable lattice models).

Infinitesimal geometry: a link between smooth and discrete theories?

Recently, for several (integrable) geometric configurations such as, e.g. surfaces of constant curvature, isothermic surfaces, triply orthogonal systems, discrete analogs have been defined, and studied. However, for most of them, the relation between the smooth and discrete theories is not very clear: for example, it is not clear whether/how the smooth theory can be obtained as a "smooth limit" from the corresponding discrete theory. In this talk, I want to raise the question whether a treatment of the smooth theory in terms of nonstandard analysis can help to clarify the relation. As a motivation, I intend to discuss an example in terms of "infinitesimal geometry".

Factorization theorems for harmonic maps of surfaces into unitary groups by singular groups actions

Can every harmonic map of a surface into a unitary group be factorized as a product of some singular group actions (called flag factors or unitons)? The problem was offered by Guest and Bergvelt. We have studied it and given an answer in the affirmative.

Compact manifolds with exceptional holonomy

In the classification of Riemannian holonomy groups, the most mysterious cases are the exceptional holonomy groups $G_2$ in 7 dimensions and Spin(7) in 8 dimensions. The first known examples of compact 7- and 8-manifolds with these holonomy groups were found by the author in 1994-5. The talk describes this construction, and discusses questions for future research. To construct compact 7-manifolds with holonomy $G_2$ we first choose a flat $G_2$-structure on the torus $T^7$, preserved by a finite group $G$ of diffeomorphisms of $T^7$. Then $T^7/G$ is an orbifold. Secondly, we resolve the singularities of $T^7/G$ to get a compact 7-manifold $M$. Finally we use analysis, and an understanding of Calabi-Yau metrics, to construct a family of metrics with holonomy $G_2$ on $M$, which converge to the singular metric on $T^7/G$. Since the original papers published in 1996, the author has made considerable improvements to the construction, found many more examples, and developed a new construction for compact 8-manifolds with holonomy Spin(7) starting from Calabi-Yau 4-orbifolds, which yields rather large Betti numbers. There will not be time to discuss all this in detail in the talk, but you can read about them in my book, D. Joyce, "Compact manifolds with special holonomy", OUP, July or August 2000.

On Kaehler-Liouville manifolds

In the paper "Memoirs, AMS, 130/619" I defined and studied two major classes of riemannian manifolds whose geodesic flows are (completely) integrable; they are called Liouville manifolds and Kaehler-Liouville manifolds respectively. Liouville manfiolds are manifolds whose geodesic flows are integrated in a similar way as those of ellipsoids, and Kaehler-Liouville manifolds are a hermitain version of Liouville manifolds. The complex projective space with the standard Kaehler metric is a typical example of the latter class.

Exactly speaking, only n first integrals are given in the definition for a complex n-dimensional K-L manifold; they are simultaneously normalizable hermitian forms on each cotangent space. However, it turns out that, under some nondegeneracy condition (called "of type (A)"), other n first integrals appear automatically as mutually commutative, infinitesimal automorphisms of the Kaehler manifold, and the geodesic flow becomes integrable. Moreover, if the manifold is compact, then those infinitesimal automorphisms generate a torus action so that the manifold becomes a toric variety.

In this lecture I will talk about the structure of compact K-L manifolds that are not necessarily of type (A). I will show that such a manifold has a bundle structure whose base is a product of one-dimensional Kaehler manifolds and whose fibers have integrable geodesic flows.

Morse complex for a real iso-spectral manifold generated by the generalized Toda lattice

We give a Morse complex for the real iso-spectral manifolds generated by the generalized Toda lattice for real split semisimple Lie algebras. This is joint work with Luis Casian.

Topology of hyperK\"ahler quotients

We study the topology of some classes of hyperK\"ahler quotients by tori and some non-abelian Lie groups. In particular we determine their cohomology rings. We also discuss its application to the topology of symplectic quotients.

The Virasoro conjecture for Gromov-Witten invariants

The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function for Gromov- Witten invariants of a projective variety is annihilated by an infinite sequence of differential operators which form a half branch of the Virasoro algebra. In the case when the underlying manifold is a point, this conjecture is equivalent to Witten's conjecture, which was proved by Kontsevich, that the corresponding generating function is a tau function of the KdV hierarchy. In this talk, I will discuss the current state of this conjecture and the proof of this conjecture for the genus 0 and (partially) genus 1 cases.

I will study the characterisation of $p$-harmonic morphisms between Riemannian manifolds, in the spirit of Fuglede-Ishihara. After a result establishing that $p$-harmonic morphisms are precisely horizontally weakly conformal $p$-harmonic maps, we compare ($2$-)harmonic morphisms and $p$-harmonic morphisms ($p> 1, p \neq 2$).

The Frenet-Serret and generalized Weierstrass relations from the Dirac equations

I will show that Dirac operator restricted to an immersed submanifold in euclidean flat space exhibits the immersed properties of the submanifold. The Frenet-Serret and generalized Weierstrass relations are identified with the Majorana representations of Dirac equations on a space curve and a conformal surface in euclidean space respectively. Further their operator determinants are the Euler-Bernoulli and the Willmore functional energy. Deformations preserving these energies are reduced to the Lax equations, or soliton equations, whose operators are the Dirac operators. I will show that the Dirac operator can be systematically defined and there is a possibility to extend our theory to the higher dimensional case.

V. S. Matveev

Integrabilities in the theory of geodesically equivalent metrics

We show that if two metrics on one closed connected manifold have the same geodesics then we can construct invariantly a set of commuting linear differential operators of second order. The Laplacian of the first metric is one of these operators.If all eigenvalues of one metric with respect to another are different at a point of the manifold then theoperators are independent and therefore the geodesic flow of any of these metrics is completely integrable, in the classical and in the quantum sense. As a corollary we have that the manifold is a ramification of the torus and that it is possible to separate the variables in the equation on eigenfunctions of the Laplacian: we can reduce this equation to a set of one-dimensional Schroedinger equations.

Rational minimal surfaces through the UP-iteration

A special class of minimal surfaces of genus zero with finite total curvature, with Enneper type ends, and without umbilics will be described and many examples of these surfaces can be produced using the UP-iteration -- a combination of Mobius transformations and Christoffel transformations -- for the Gauss maps of these surfaces. The proof that the UP-iteration yields such surfaces uses properties of the Schwarzian derivative. Recent work with M. Umehara has provided the analogue for CMC1 surfaces in hyperbolic 3-space.

Harmonic tori and their spectral data.

It is known that each non-isotropic harmonic torus in a sphere or complex projective space is determined by data on an associated compact Riemann surface, called the spectral curve. I will explain a concise way of looking at this relationship and how it generalises to give harmonic maps into Grassmannians and unitary groups. The major unsolved problem in this field is to provide an effective method for identifying which spectral data provides tori. It is similar to the problem of finding periodic solutions of integrable systems and I will discuss how these similarities might help understand certain nice classes of solutions.

Combinatorics of coinvariants

In the study of solvable lattice models many interesting combinatorial identities have been found. Among those are the generalizations of the famous Rogers-Ramanujan identities. Several years ago, McCoy and his collaborators proposed to give names to each side of such identities: Fermionic and bosonic character formulas. In this talk, we give a construction of Fermionic character formulas for certain spaces of $\widehat{sl}_2$ co-invariants by using representation theory. This is a joint work with B. Feigin, R. Kedem, S. Loktev and E. Mukhin.

Integrability and homogeneity of hypersurfaces

We interpret the homogeneity of hypersurfaces in the euclidean sphere as the integrability of a certain Lax equation, satisfied by the shape operators of the focal submanifolds.

(1,2)-symplectic structures on flag manifolds

By using moving frames and direct digraphs, we study invariant $(1,2)$ symplectic structures on complex flag manifolds. Let $F$ be a flag manifold with height $k-1$. We show that there is a $k$-dimensional family of invariant $(1,2)$-symplectic metrics of any parabolic structure on $F$. We also prove that any invariant almost complex structure $J$ on $F$ with height $3$ admits an invariant $(1,2)$ symplectic metric if and only if $J$ is parabolic or integrable.

As a method of providing a unified viewpoint for most integrable systems, it is well known that the 4-dimensional ASD equation plays an important role through symmetry reductions. In this context, the twistor theory naturally explains the algebraic geometrical features which the integrable system posseses (eg Nahm's equation).

Both the ASD equation and the twistor theory can be generalized on hyperK\"ahler and quaternion-K\"ahler manifolds. The dimensional reduction gives us the (generalized) Nahm's equation with more Higgs fields than the usual one from a generalized ASD equation on the quaternionic projective space.

By extending an ASD connection to a higher dimensional object, we obtain a non-trivial zero locus of a section which satisfies a linear field equation, the twistor equation. In my talk, it is shown that the zero locus of such a section -the twistor section- is a quaternionic submanifold. The Penrose transform converts a twistor section into a holomorphic section on a holomorphic bundle over the twistor space. In general, the zero locus of the corresponding holomorphic section is not the same as the inverse image of the zero locus of a twistor section. However, under the reality condition, these are the same as each other, and so the zero locus of the holomorphic section is the twistor space of a quaternionic submanifold. On the other hand, some cohomological conditions for holomorphic bundles over the twistor space yields the connectedness of the zero locus of a twistor section.

Some examples of twistor sections will be exhibited on the quaternion symmetric spaces, the Wolf spaces. Since the twistor space of the Wolf spaces are also homogeneous, the Bott-Borel-Weil theory makes them easier to specify a twistor section and to describe its zero locus. It is one of the typical examples that a quaternionic submanifold $CP^2$ of $G_2/SO(4)$ is obtainable as the zero locus of a twistor section.

As an application, we will discuss the ideal ASD connection, which corresponds to a boundary point of the moduli space of ASD connections, and so has a singular set. It will be shown in some examples that the Poincar\'e dual of the singular set is the even degree Chern class of a vector bundle on which the ideal ASD connection is defined.

The author obtained most of these results during his stay at Oxford. He gratefully acknowledges Simon Salamon for much valuable advice.

Integrable systems and cohomologies of affine hyperelliptic Jacobi varieties

We will discuss the integrable system associated with the family of hyperelliptic curves. The space of functions on the phase space modulo the action of the integrals of motion is studied. The main ingredient in the study is the character of the affine ring of the affine Jacobi variety, the complement of the theta divisor. Using the matrix construction of the affine hyperelliptic Jacobi variety due to Mumford, we explicitly determine the character. By decomposing the character we make conjectures on the cohomology groups of affine Jacobi varieties and on the space of abelian functions. This is a joint work with F. Smirnov.

Birational realization of Weyl groups and discrete integrable systems

We will present a general method to realize the Weyl group defined by an arbitrary root system (or generalized Cartan matrix) as a group of birational canonical transformations. In the case of affine Weyl groups, this construction provides a class of discrete integrable systems which can be regarded as the symmetry of certain similarity reductions of the modified Drinfeld-Sokolov hierarchies. This work has its origin in the study of Backlund transformations of the Painleve equation.

On harmonic maps into symmetric spaces and gauge-theoretic approach

We shall discuss harmonic maps of Riemann surfaces, more generally pluriharmonic maps of complex manifolds, into compact symmetric spaces, and their moduli spaces from the viewpoint of the gauge-thoeretic equations. We shall mention about some recent results of joints works with Mariko Hidano Mukai (PD, JSPS and TMU) and Seiichi Udagawa (Nihon University), and some related problems.

Isomonodromy Deformations and Twistor theory

Painleve functions are transcendental in general but there are special solutions of Painleve equations, which reduce to linear differential equations. Such particular solutions appear on walls of the action of affine Weyl groups.

We will study particular solutions of degenerate Schlesinger equations. Particular solutions of Schlesinger equations are represented by Aomoto-Gelfand hypergeometric functions. Aomoto-Gelfand hypergeometric functions can be considered as twistor transformation of line bundles and Schlesinger equations can be considered as twistor transformation of vector bundles. In this sense, Painleve functions are non-abelian analogues of hypergeometric functions.

The Painleve-Garnier Systems

The Garnier Systems are complete integrable systems, defined by holonomic deformation of linear ordinary differential equations. Regarding them as extension of the Painleve equations, we consider the Hamiltonian structure of these systems and study relationships among them.

Simple singularities and symplectic fillings

For a given contact manifold, there may exist compact symplectic manifolds bounding the contact manifold with a certain compatibility condition (roughly convexity of the boundary with respect to the symplectic structure). Such a compact symplectic manifold is called a symplectic filling of the contact manifold. The link of an isolated singularity of algebraic variety is a typical example of contact manifolds. In this case, there are two "natural" symplectic fillings. Namely, the Milnor fiber and the minimal resolution. They have different topology, in general. However, if the isolated singularity is so-called simple singularity in complex dimension 2, they are diffeomorphic (Brieskorn). We will discuss such uniqueness of diffeomorphism type of symplectic fillings of links of simple singularities. This is a joint work with Hiroshi Ohta (Nagoya University).

Complex structures on some Stiefel manifolds

We construct a complex structure on the total space of an induced Hopf $S^3$-bundle over a Sasakian manifold. We apply this construction to the Stiefel manifolds $V_2(C^n)$, $V_4(R^n)$ and to some "special Stiefel manifolds" related to Cayley numbers ($G_2$, $Spin(7)/Sp(1)$).

Symmetric submanifolds of symmetric spaces

Constructions are given of new examples of symmetric non-totally geodesic submanifolds in irreducible symmetric spaces of non-compact type and rank greater than one.

Quantum cohomology of infinite dimensional flag manifolds and the periodic Toda lattice

We show that the result of Givental-Kim on the relation between the quantum cohomology of flag manifolds and the open Toda lattice can be generalized to infinite dimesional flag manifolds and the periodic Toda lattice. This is joint work with M. Guest.

An efficient method for solving initial value problems for weakly nonlinear wave equations

Wineberg, McGrath, Gabl, and Scott developed a pseudospectral method for integrating the KdV initial value problem numerically, and we have extended their algorithm to solve initial value problems for a wide class of evolution equations that are "weakly nonlinear'' in a sense we will make precise. This class includes the other classical soliton equations (SGE, and NLS) and many more besides. As well as being very simple to implement, this method exhibits remarkable speed and stability, making it an ideal method for visualization tools such as 3D-Filmstrip, where it permits real-time experimentation with how solitons interact or how general solutions decompose into solitons. We will demonstrate such uses in the lecture, and in addition we will analyze the structure of the algorithm, showing the reasons behind its nice behavior. Finally we will describe the fixed point theorem we have found that proves that the pseudospectral stepping algorithm converges to the actual solution.

Submanifolds associated to Grassmannians

Using Grassmannian systems, we study immersions of space forms into space forms, constant mean curvature surfaces in the 3-dimesional Euclidean space. Also, as with Backlund and Darboux transforms, we give a method to produce submanifolds of the same type from a given one.

Quasi-Einstein K\"ahler Metrics

We write an ansatz for quasi-Einstein K\"ahler metrics and construct new complete examples. Moreover, we construct new compact generalized quasi-Einstein K\"ahler metrics on some ruled surfaces, including some of Guan's examples as special cases.

Quaternionic holomorphic geometry and energy estimates

I will apply quaternionic holomorphic Riemann surface theory to estimate the harmonic map and Willmore energies.

Quaternionic Pluecker formula and the differential geometry of surfaces

The classical Pluecker formula for counting singularities of complex algebraic curves has a quaternionic analog that can be used to estimate the area of minimal tori in the three sphere, eigenvalues of the Dirac operator on Riemann surfaces, and other geometric quantities.

Hamiltonian stationary Lagrangian surfaces and integrable systems-2

We study Lagrangian surfaces in homogeneous four-dimensional symplectic Kaehler manifolds which are critical points of the area functional for Hamiltonian vector field deformations. This problem turns to be completely integrable. It leads to various descriptions with a dramatic simplification for surfaces in Euclidean space.

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) $1$ in $\R^{n}$ (Delaunay unduloids). When$n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than $1$ in hyperbolic space.

Discrete conjugate net of D-branes

The discrete conjugate net, which was discussed by Doliwa and Santini, is shown to appear as location of D-branes in the string theory.

Harmonic cohomology groups on compact symplectic nilmanifolds

Let $(M, \omega)$ be a symplectic manifold. J. Brylinski introduced the notion of symplectic harmonic forms. He defined the star operator $*:{\Omega}^{k}(M) \rightarrow {\Omega}^{k-1}(M)$, where ${\Omega}^{k}(M)$ is the space of all $k$-forms on $M$, and operator $d^{*} = (-1)^{k}*d*$ on ${\Omega}^{k}(M)$. A form $\alpha$ is called a (symplectic) harmonic form if it satisfies $d\alpha = d^{*}\alpha = 0$. Let ${\Cal H}^{k}(M)$ denote the space of all harmonic $k$-form and define the symplectic harmonic $k$-cohomology group $H^{k}_{hr}(M) $ of the symplectic manifold $(M, \omega)$ by ${\Cal H^{k}(M)}/(B^{k}(M) \cap {\Cal H}^{k}(M))$, as a subspace of the de Rham cohomology group $H^{k}_{DR}(M)$. We denote by $L_{\omega} : {\Omega}^{k}(M) \rightarrow {\Omega}^{k+2}(M)$ the multiplication by $\omega$ and by $L_{[\omega]} : H^{k}(M) \rightarrow H^{k+2}(M)$ the induced homomorphism in the de Rham cohomology. We say that a symplectic manifold $(M^{2m}, \omega)$ has the Hard Lefschetz property if, for every $k \leq m$, the homomorphism $L_{[\omega]}^k : H^{m-k}_{DR}(M) \rightarrow H^{m+k}_{DR}(M)$ is surjective.�@O. Mathieu proved that for a symplectic manifold $(M^{2m}, \omega)$, any de Rham cohomology class contains a harmonic cocycle if and only if $(M^{2m}, \omega)$ has the Hard Lefschetz property. Dong Yan gave a simpler, more direct, proof of Mathieu's Theorem and studied harmonic cohomology groups $H^{k}_{hr}(M)$ of compact symplectic 4-manifolds. In particular, he proved that $L_{\omega}^{k}:{\Cal H}^{m-k}(M) \rightarrow {\Cal H}^{m+k}(M)$ is an isomorphism.

In this talk, based on works with Takumi Yamada, we consider compact symplectic nilmanifolds and discuss a question raised by B. Khesin and D. McDuff: On which compact manifold $M$ does there exist a family $\omega^{}_{t}$ of symplectic forms such that the dimension of $H^{k}_{hr}(M)$ varies ?

Invariant Hermitian metrics

A selection of examples and results concerning some special Hermitian metrics on Lie groups of real dimension 4 and 6, and related integrable structures.

Nongeneric flows in the full Kostant-Toda lattice

This paper studies singular flows on the generic symplectic leaves in the full Kostant-Toda lattice of $sl(n, {\bf C})$ and the flows on arbitrary nongeneric leaves in low-dimensional examples. The method involves embedding a level set of the integrals into the flag manifold of $Sl(n,{\bf C})$ or a product of partial flag manifolds. Here generic flows correspond to generic orbits of the diagonal torus $({\bf C}^*)^{n-1}$. In nongeneric flows, either the torus degenerates into $({\bf C}^*)^{r-1} \times {\bf C}^{n-r}$ or the flows of the diagonal torus become nongeneric. The latter singularity is reflected in splittings of momentum polytopes of flag manifolds along interior faces.

Bonnet Pairs in the Three Dimensional Sphere

If there are exactly two non-congruent immersed surfaces with the same metric and mean curvature function, one speaks of a Bonnet pair. It is shown that Bonnet pairs in the three dimensional sphere are described by an integrable system, a Lax representation of which is presented. Simple examples are discussed and the connection with isothermic surfaces is clarified.

Frobenius submanifolds

The notion of a Frobenius submanifold - a submanifold of a Frobenius manifold which is itself a Frobenius manifold with respect to structures induced from the original Frobenius manifold - is studied. Two dimensional submanifolds are particularly simple. More generally, sufficient conditions are given for a submanifold to be a so-called natural Frobenius submanifold. These ideas are illustrated using examples of Frobenius manifolds constructed from Coxeter groups, and for the Frobenius manifolds governing the quantum cohomology of CP^2 and CP^1 * CP^1.

Involutive structures on the tangent bundle of symmetric spaces

Geodesic flow invariant complex structures are important since they provide a way of quantizing the energy function on the tangent bundle of such manifolds. For compact rank one symmetric spaces this was studied by Rawnsley, Furutani, Tanaka and Yoshizawa. With the help of the so called adapted complex strutures we shall examine the higher rank symmetric spaces. Here the situation is much more complicated. One can only get a geodesic flow invariant complex polarization.

Subspaces in the category of symmetric spaces

This talk is based on collaboration with Tadashi Nagano.

Let $M$ be a compact symmetric space. Each connected component of the fixed point set of $s_o$ is called {\it a polar} of $o$ in $M$ and denoted by $M^+_{(p)}$ if it contains a point $p$. The connected component of the fixed point set of $s_p \circ s_o$ which contains $p$ is called {\it the meridian} to $M^+_{(p)}$ at $p$, denoted by $M^-_{(p)}$. A polar and the meridian to it are subspaces in $M$ which play fundamental roles in the category of symmetric spaces. In fact, a compact connected symmetric space is globally determined by any one pair of a polar and the meridian to it. When a polar $M^+_{(p)}$ consists of a single point $p$, $p$ is called {\it a pole} of $o$ in $M$ if $p$ is not $o$.

Let $o, p$ and $q$ be three points in $M$. Triplets $\{o, p, q\}$ is called {\it a regular triplet} in $M$ if $s_o \circ s_p \circ s_q = \mbox{id}_M$. For example, the real, complex, quaternion, or Cayley projective plane has a regular triplet. In fact, the polar $M^+_{(p)}$ of a point $o$ in $M$ is a projective line and taking $q$, the pole of $p$ in $M^+_{(p)}$, we have a regular triplet $\{o, p, q\}$. Then $o, p$ and $q$ are vertices of ``a right equilateral triangle'' each of whose sides is a projective line.

If $M=G/K$ has a regular triplet $\{o, p, q\}$, the Lie algebra $\mathfrak{g}$ of $G$ has an orthogonal direct decomposition $\mathfrak{g}=\mathfrak{k}_+ + \mathfrak{k}_- + \mathfrak{m}_+ + \mathfrak{m}_-$ with respect to $s_o$ and $s_p$, and $\mathfrak{k}_- \cong \mathfrak{m}_+ \cong \mathfrak{m}_-$ as $\mathfrak{k}_+$-modules. When $M$ has no pole, we have the following fact: if a polar $M^+_{(p)}$ of $o$ in $M$ has a pole $q$ in it, then (i) $M^+_{(p)}$ is isomorphic with $M^-_{(p)}$ and (ii) $\{o, p, q\}$ is a regular triplet in $M$. To prove this we use the following result: for a pair $(M^+_{(p)}, M^-_{(p)})$ of a polar $M^+_{(p)}=K/K_+$ and its meridian $M^-_{(p)}=G_-/K_+$ of the origin $o_M$ in a compact symmetric space $M=G/K$, there is a subspace $S$ of a symmetric space $N=G/G_-$ containing the origin $o_N$ of $N$ which satisfies that (i) there is the orthogonal subspace $S^\perp$ to $S$ at $o_N$ and (ii) $S \cong S^\perp$ and they are locally isomorphic with $M^+_{(p)}$.

Periodicity conditions of harmonic maps associated to spectral data

McIntosh proved that every non-isotropic harmonic torus in a complex projective space corresponds to a map constructed from a triplet $(X, \pi, \mathcal{L})$, consisting of an auxiliary algebraic curve $X$, and a rational function $\pi$ and a line bundle $\mathcal{L}$ on $X$. Such atriplet is called spectral data. Thus McIntosh realized the moduli space of non-isotropic harmonic tori in complex projective spaces as a subset of the moduli space of these spectral data.

Therefore it seems natural to ask the following: {\it Which spectral data corresponds to a harmonic torus in a complex projective space?}

In this talk, we give a partial answer to this problem. More precisely, we prove a criterion on the periodicity of harmonic maps constructed from the spectral data whose spectral curves are smooth rational or elliptic curves. In order to get the result, we compute the kernels of certain homomorphisms into generalized Jacobians introduced by McIntosh.

References:

[1] I. McIntosh, A construction of all non-isotropic harmonic tori in complex projective space, Internat.\ J.\ Math.\ 6 (1995), 831-879.

[2] I. McIntosh, Two remarks on the construction of harmonic tori in ${\Bbb C}P^{n}$, Internat.\ J.\ Math.\ 7 (1996), 515-520.

[3] I. McIntosh, Harmonic tori and generalised Jacobi varieties, math.DG/9906076.

[4] T. Taniguchi, Non-isotropic harmonic tori in complex projective spaces and configurations of points on rational or elliptic curves, T\^{o}hoku Math. J. (To appear).

Hermann actions of unitary groups and integral geometry in complex projective spaces

The natural action of the unitary group on the Grassmann manifold consisting of subspaces in a real vector space of even dimension is a Hermann action. We can describe a section of this action explicitly and obtain a generalization of K\"ahler angle of a subspace. We formulate a Poincar\'e formula of submanifolds in the complex projective space, using a generalized Poincar\'e formula obtained by Howard and this generalization of K\"ahler angle. In some cases of low dimensions we can show Poincar\'e formulas in more explicit ways.

Higher-dimensional parallel transports

For a smooth path in a manifold X, a connection on a line bundle over X gives an isomorphism between the fibers called parallel transport along the path. In this talk, viewing a path as a map from a 1-dimensional manifold with boundary to X and a line bundle with connection as a 1-dimensional smooth Deligne cocycle of X, we generalize such a concept to maps from a higher-dimensional manifold with boundary to X and a higher-dimensional smooth Deligne cocycle, using transgression maps on smooth Deligne cochains.

Submanifolds arising from soliton equations

There is a natural hierarchy of commuting soliton equations associated to each symmetric space. I will explain how submanifold geometries arise from these equations. If the symmetric space has rank k, then the submanifolds have dimension k. I will use Dick Palais' 3D-filmstrip to show some of the surfaces corresponding to specific rank 2 symmetric spaces.

Integrability criterion of geodesical equivalence

We prove that the Riemannian metrics $g$ and $\bar g$ (given in general position) are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric $g$. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent metrics. We show that the metrics of the standard ellipsoid and the Poissin sphere lie in such a hierarchy.

Surface of constant mean curvature 1 in hyperbolic 3-space

Recent developments in the study of constant mean curvature $1$ surfaces in hyperbolic $3$-space $H^3$ (of constant sectional curvature $-1$) have led to many recently-discovered examples of such surfaces, and it is now well-known that CMC $1$ surfaces in $H^3$ share quite similar properties with minimal surfaces in Euclidean $3$-space $\R^3$.

The total absolute curvature of a complete minimal surface in $\R^3$ is a $4\pi$-multiple of a nonnegative integer and is equal to the area of the Gauss image of the surface. All such surfaces with finite total absolute curvature less than or equal to $8\pi$ have been classified (Lopez~\cite{Lopez}). Here we consider the corresponding problem for CMC $1$ surfaces in $H^3$.

Unlike the case of minimal surfaces in $\R^3$, CMC $1$ surfaces in $H^3$ have two Gauss maps, the hyperbolic Gauss map $G$ and the secondary Gauss map $g$. The total absolute curvature of CMC $1$ surfaces in $H^3$ is equal to the area of the image of the secondary Gauss map $g$, but since $g$ might not be single-valued on the surface, the total absolute curvature might not be a $4\pi$-multiple of an integer. The hyperbolic Gauss map $G$, on the other hand, does not relate to the total absolute curvature of the surface directly, but it has much clearer geometric meaning, namely the image $G(p)$ lies in the ideal boundary $S^2$ of the hyperbolic space at the point corresponding to the end of the normal geodesic emanating from the point $p$ on the surface. Therefore, the hyperbolic Gauss map $G$ is single-valued on the surface.

The area of the image of the hyperbolic Gauss map $G$ is called the dual total curvature, which is equal to the total curvature of the dual CMC-1 surface. In particular, the dual total absolute curvature is always a $4\pi$-multiple of an integer. Though the total absolute curvature of CMC $1$ surfaces satisfies only the Cohn-Vossen inequality, the dual total absolute curvature has a much stronger lower bound, which is an analogue of the Osserman inequality for minimal surfaces (cf.~\cite{uy5,Yu2}).

Constructing and classifying \cmcone{} surfaces in $H^3$ with low total absolute curvature turns out to be more difficult and subtle than Lopez's classification, since the Bryant representation formula, which is an analogy of the Weierstrass representation formula, is not formulated by using line integration, but rather uses parallel transport along a path in the non-commutative group $\SL(2,\C)$.

The purpose of this talk is to give a method to construct new surfaces and to list the possibilities of CMC $1$ surfaces in $H^3$ with total absolute curvature, or dual total absolute curvature less than or equal to $8\pi$, which is a joint work with Wayne Rossman.

Moreover, we shall mention a generalization of CMC-1 surface theory to certain non-compact type symmetric spaces, which is a joint work with Masatoshi Kokubu and Masarou Takahashi.

\bibitem{Lopez} F.~J.~Lopez, {\itshape The classification of complete minimal surfaces with total curvature greater than $-12\pi$}, Trans.~Amer.~Math.~Soc. {\bfseries 334} (1992), 49--74.

\bibitem{uy5} M.~Umehara and K.~Yamada, {\itshape A duality on \cmcone{} surface in the hyperbolic $3$-space and a hyperbolic analogue of the Osserman Inequality}, Tsukuba J. Math. {\bfseries 21} (1997), 229-237.

\bibitem{Yu2} Z.~Yu, {\itshape The inverse surface and the Osserman Inequality}, Tsukuba J. Math. {\bfseries 22} (1998), 575--588.

Geometry of the singularities of integrable Schroedinger operators

Let M be a Riemannian manifold, L_0 be the Laplace-Beltrami operator on M, L = L_0 + u(x) a Schroedinger operator on M with potential u(x). Consider the following problem: for which u(x) does there exist a differential operator D on M intertwining L and L_0: L D = D L_0 ? An example of such a u(x) is given by the well-known Calogero-Moser potential in Euclidean space. This potential has the singularities located on a Coxeter configuration of hyperplanes. It turns out that the geometry of the singularities must be very special also in the general case. More precisely, one can show that the singular hypersurface of any operator L satisfying an intertwining relation must be totally geodesic. For the Euclidean space this has been proven by Yu. Berest and the speaker, and in the general case by M.Feigin. The description of the corresponding configurations of totally geodesic hypersurfaces in the spaces of constant curvature is a very interesting problem, which is still open even in the Euclidean case. Some results in this direction found by O.Chalykh, M.Feigin and the speaker will be discussed in the talk.

Conservation Law and Global Existence Theorem for Geometric Schroedinger Equations

We will discuss the Conservation Law for Schroedinger flows for maps into Hermitian symmetric spaces. As an application of this law, we prove the global existence of Schroedinger flows for maps from the circle into Hermitian symmetric spaces.

Jacobi fields along harmonic maps (work with L. Lemaire)

In [1], following topological work of Crawford, we showed that the components of the space of harmonic maps from the $2$-sphere $S^2$, to complex projective 2-space $CP^2$ are smooth submanifolds of the space of all, say, $C^2$ maps from $S^2$ to $CP^2$, of given (finite) dimensions depending on the energy and degree of the maps In this lecture, we shall show that any Jacobi field along a harmonic map from $S^2$ to $CP^2$ is integrable in the sense that it arises from a variation through harmonic maps. (Note that Jacobi fields along harmonic maps are not always integrable, for example, the normal Jacobi field to a closed geodesic around the flat waist of a suitable surface of revolution.)

This result answers a question posed by L. Simon whilst giving his survey lectures at the first MSJ International Research Institute and has implications for the behaviour of a harmonic map near a singularity, see [2, p. 131], as well as identifying the tangent space to the space of harmonic $2$-spheres in $CP^2$.

[1] Lemaire, L. and Wood, J.C., On the space of harmonic $2$- spheres in ${C}{\rm P}^2$, Internat. J. Math. {\bf 7} (1996), 211-- 225.

[2] Simon, L. Theorems on the regularity and singularity of minimal surfaces and harmonic maps. Geometry and global analysis (Sendai, 1993), 111--145, Tohoku Univ., Sendai, 1993.

Global Poisson Group Actions of the ZS-AKNS Flows

The global Poisson group actions and structures of the $U(2)$- and $U(1,1)$-ZS-AKNS flows will be investigated using the inverse scattering method.

Construction of constant mean curvature trinoids from holomorphic potentials

Using a Weirstrass type representation, we construct a 3-parameter family of constant mean curvature trinoids in space from holomorphic potentials on the surfaces. The ends of these trinoids are asymptotically Delaunay. Moreover, an end is embedded if the corresponding asymptotic Delaunay surface is.