Country：Japan

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Country：Japan

researchmap

Country：Japan

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Country：Japan

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Grant amount：\17550000 （ Direct Cost: \13500000 、 Indirect Cost：\4050000 ）

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Grant amount：\32890000 （ Direct Cost: \25300000 、 Indirect Cost：\7590000 ）

Kossowski metrics characterize the singularities appeared on wave fronts. We (the head investigator and his research group) investigated global properties of them. In particular, we showed that Each Kossowski metric induces a unique coherent tangent bundle. Using this, two Gauss-Bonnet type formulas were obtained. Also, we showed the existence of four distinct cuspidal edges along a given space curve with the same first fundamental form in Euclidean 3-space. On the other hand, using analytic extension formula along fold singularities, we construct several families of real analytic entire zero-mean curvature graphs of mixed type in Lorentz-Minkowski 3-space.

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Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）

A class of maximal surfaces in Lorentz-Minkowski 3-space, named "Kobayashi surfaces" is introduced. A surface in this class can be extended to a zero-mean curvature surfaces which changes causal types from space-like to time-like. Existence of infinitely many surfaces among this class which are graphs of functions over the space-like plane. On the other hand, the first example of a zero-mean curvature which contains a light-like line and changes causal types across the line is obtained. Such a property, called the light-like line theorem, are generalized for wider class of surfaces.
For a surface in Lorentzian 3-manifold which changes its causal type from space-like to time-like at a light-like point, it is shown that the mean curvature function converges to zero at the light-like point. In particuler, it is shown that a non-zero constant mean curvature surface cannot change its causal type.

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Grant amount：\12350000 （ Direct Cost: \9500000 、 Indirect Cost：\2850000 ）

By making use of Cheng-Yang recursion formula, we give optimal estimates for lower bounds of eigenvalues of Laplacian on a bounded domain in complete Riemannian manifolds. Our method is original. According to this result, a difficult problem proposed by I. Chavel is solved. Furthermore, we find an obstruction on minimal immersions from complete Riemannian manifolds into Euclidean spaces in the view of eigenvalues of Laplacian. Geometry of fronts with singularities has been studied. Gauss-Bonnet theorem on fronts is proved. By improving the generalized maximum principle of Omori-Yau, important results on classification of complete self-shrinkers of the mean curvature flow are obtained. Eigenvalues of Laplacian on compact Alexandrov spaces are studied and important progresses are obtained.

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Grant amount：\13780000 （ Direct Cost: \10600000 、 Indirect Cost：\3180000 ）

Isoparametric hypersurfaces with 6 principal curvatures with multiplicity 2 are shown to be homogeneous, which solves one of Yau's problems. As for 4 principal curvature case, we gave a description by using the moment map of spin actions. Transnormal systems are investigated in details.
We show the non-existence of L2 harmonic 1-form on a complete non-compact stable minimal Lagrangian submanifolds in a Kaheler manifold with positive Ricci curvature. Then the number of non-parabolic ends is less than two, and in the surface case, the genus should vanish. The Floer theory on the intersection of a Lagrangian submanifold with its Hamiltonian deformation is investigated. The Gauss images of isoparametric hypersurfaces in the sphere are Lagrangian submanifolds of complex hyperquadric, and in this case, we show that if the multiplicities of the principal curvatures are bigger than 1, then they are Hamiltonian non-displaceable,

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Grant amount：\30940000 （ Direct Cost: \23800000 、 Indirect Cost：\7140000 ）

Using the concept of coherent tangent bundles, we (the head investigator and the research group) found four new Gauss-Bonnet type formulas for closed surfaces with singularities in Euclidean 3-space. Using the fact that a spacelike maximal surface with fold singularities has an analytic extension across those singularities, we showed that the analytic extensions of the triply-periodic Schwarz D type maximal surfaces are all embedded. In a joint work with Yoshihisa Kitagawa, we proved that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Moreover, we obtained some interesting results for plane curves. For example, a simplification of the proof of Bol's conjecture on sextactic points was discovered.

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Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）

We studied surfaces admitting singularities in some kind of three-dimensional manifolds of constant curvature, requiring them to have good properties from the differential-geometric viewpoint. (Note that non-Euclidean space of constant curvature have interesting features beyond our common sense.) Concerning linear Weingarten surfaces in hyperbolic space, we had a global representation formula, criterion for the shape of singularities, and a result on the orientability and the co-orientability. Concerning CMC-1 faces in de Sitter space and maxfaces in Lorentz-Minkowski space, we had results on the orientability and the co-orientability. At the same time, the classification of CMC-1 faces having two ends were obtained, and the classification of maxfaces having three ends were obtained.

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Grant amount：\3110000 （ Direct Cost: \2600000 、 Indirect Cost：\510000 ）

We studied surfaces with constant mean curvature and surfaces with constant anisotropic mean curvature with free or fixed boundary. We obtained results about existence, uniqueness, geometric properties of solutions or stable solutions. Also, we obtained sufficient conditions for the existence of bifurcationand criterion of the stability for each surface in the bifurcation branch. Moreover,by removing the convexity assumption for the anisotropic surface energy, we studied uniformly a large class of surfaces including constant mean curvature surfaces in the Lorentz-Minkowski space and obtained a new uniqueness theorem and examples.

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Grant amount：\15340000 （ Direct Cost: \11800000 、 Indirect Cost：\3540000 ）

We investigated, Weierstrass-type representation formula, global properties of several classes of surfaces with singularities, such as flat surfaces in hyperbolic 3-space, maximal surfaces in Minkowski 3-space, CMC-1 surfaces in de Sitter 3-space, and improper affine sphere in affine 3-space, and obtained a charctreization of completeness, Osserman-type inequalis etc.
In addition, flat trinoids in hyperbolic space and CMC-1 2-noids in de Sitter 3-space are classified. On the other hand, as a basic tool of differential geometry of wave front, we introduced a notion of "sinular curvature" and investigated a rdelationship between singular curvature and behavior of Gaussian curvature. As a result, we obtained Gauss-Bonnet type formula for wave fronts. Moreover, as an intrinsic formulation of wave fronts, we introduced a notion of "coherent tangent bundles" and gave an application of their duality.

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Grant amount：\27560000 （ Direct Cost: \21200000 、 Indirect Cost：\6360000 ）

We classified almost all isoparametric hypersurface, and characterize them in terms of the moment map, which proves the evidence of a relation with integrable systems. A basic theory of surfaces with singularities, and a new method using the Legendre map have been established. Via the Riemann-Hilbert correspondence, the dynamical system of Painleve equations is investigated, and the view point of the chaos has been developed. The modularity of higher genus Gromov-Witten and the mirror symmetry are discussed. A surface with potential appeared in quantum cohomology is constructed, which contributes to the tt* geometry.

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Grant amount：\23270000 （ Direct Cost: \17900000 、 Indirect Cost：\5370000 ）

This research was focused on the geometry of curves and surfaces with singularities. We gave a useful criterion for A_k singular points on hypersurfaces, and applied it to the study of inflection points on hypersurfaces. This riterion enabled us to define A_k singularities of wave front without assuming the existence of an ambient space. In fact, we defined the notion "coherent tangent bundle", giving an intrinsic formulation for wave fronts and several other applications. Moreover, we investigated maximal surfaces in Lorentz-Minkowski space and constant mean curvature surfaces in de Sitter space, and constructed several interesting new examples with singularities but still having certain kind of completeness.
Additionally, the head investigator and coinvestigators held several workshops (both domestic and international), which related in many fruitful discussions with geometers studying relating fields.

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Grant amount：\4080000 （ Direct Cost: \3300000 、 Indirect Cost：\780000 ）

In this research, we studied classical differential geometry from modern viewpoints, such as of the theory of integral systems and of the theory of singularities ; we obtained results on various fields of classical differential geometry and their applications, in particular, the motions of curves associated with integrable systems, explicit construction and the classification of conformally flat hypersurfaces of four-dimensional space forms, real hypersurfaces of complex space forms, surfaces of three-dimensional spaces, affine differential geometry and its applications to Hessian geometry and information geometry, and so on.

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Grant amount：\8740000 （ Direct Cost: \7300000 、 Indirect Cost：\1440000 ）

Properties of certain classes of surfaces with singularities are investigated with Weirstrass-type representation formula. For example, global behavior of flat fronts, and behavior of singularities of maximal surfaces in Lorentz-Minkowski 3-space and mean curvature one surfaces in de Sitter 3-space are investigated.
On the other hand, as a general theory of differential geometry of singularities, a notion of singular curvature of the singular points of wave fronts is defined, and Gauss-Bonnet type formulas are obtained.

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Grant amount：\6490000 （ Direct Cost: \5800000 、 Indirect Cost：\690000 ）

In 2005, we focus our attention on submanifolds which appear as singular sets of ideal instantons. Those are zero loci of the twistor sections satisfying linear equations which are linearization of the (higher dimensional instanton equations. Moreover, we construct embeddings of the Wolf spares into Grassmannian_, which turn out to be minimal embeddings. We also obtain vanishing theorems for cohomology groups.
In 2006, we succeeded to find some relations between harmonic mappings into Grasmannians and the Yang-Mills connections, which are essential and important steps to our subject. We obtain a condition for a map of a Riemannian manifold into Grassmannian to be a harmonic map. We use this condition to obtain the classification of harmonic maps with constant energy density from holonomy irreducible homogeneous manifold s into Grassmannian manifolds. In addition, we can show that a vector bundles on a real Grassmannian manifold with some topological type admits a unique ASD connection up to gauge equivalence.
In 2007, we consider the cases that a harmonic map into Grassmannian is a totally geodesic one. As a result, we obtain the classification of totally geodesic immersions of irreducible type. In this classification, we obtain an integral formula which indicates the dimension of Grassmannian, which is the target space of the mapping. In the case of the complex projective line, we can show that an indecomposable totally geodesic immersion is an totally geodesic immersion of the irreducible type. To obtain the result, we use the above characterization of a harmonic map and construct a variant of the spherical function theory on homogeneous vector bundles. This implies that we can classify all totally geodesic immersions of complex projective line into Grassmannians. We develop an analogue of the "geometry of the twistor sections" on symmetric spaces of compact type. This gives us pairs of totally geodesic submanifolds on almost symmetric spaces of compact type. These pairs are intimately related to vector bundles and sections of them. Indeed, we can construct a function using a section, which is an isoparametric function on every Grassmann manifold. This function gives a family of submanifolds as level sets. We can find one and only minimal submanifold in this family.

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Grant amount：\34970000 （ Direct Cost: \26900000 、 Indirect Cost：\8070000 ）

Miyaoka gives a new proof for the Dorfmeister Neher classification theorem on isoparametric hypersurfaces, and as applications of hypersurface geometry, clarifies the topological structure of the anti-self-dual bundle of complex projective plane and complete austere submanifolds, constructs Ricci flat metrics, special Lagrangian submanifolds. She also gets twister fibrations from the geometry of G2 orbits. Iwasaki connects the algebraic formulation of Painleve IV with the ergodic theory of birational maps of algebraic surfaces via Riemann-Hilbert correspondence, and shows the chaotic behavior of non-linear monodoromy. Kajiwara applies the theoretic formulation of the Painleve systems and constructs the determinant formula of the hypergeometric solutions of q-Painleve, and relates it with the solutions of the associate linear problems. Nakayashiki characterizes the coefficients of the series of sigma function by those of defining functions of the algebraic curves. Nagatomo obtains an essential relation between harmonic maps and the Yang-Mills connections, and generalizes Takahashi's theorem, de Carom-Wallach's theorem, and constructs harmonic maps from quaternion Kaehler manifold to Grassmannian manifolds. Yamada-Umehara-Rossman classify the behavior of the ends of complete flat fronts in the hyperbolic 3-space. Fujioka studies integrability and periodicity of the motion of curves in complex hyperbolics which depend on Burger's equation and have descritization. Ishikawa classifies singularities of inproper affine surfaces and surfaces with constant Gauss curvature, and their dual surfaces. He also clarifies moduli of the singularities, and obtains a relation between plane curves and their Legendle curves. Udagawa classifies compact isotropic submanifolds with parallel mean curvature vector wit the sectional curvature. Tamaru proves a fixed point theorem for cohomogeneity one action corresponding to homogeneous hypersurfaces in symmetric spaces of non-compact type. Matsuura studies a development of plane curves depending on KdV equation w..r.t. discrete time. Ikeda studies equi-energy surfaces of characteristic manifod of Whittaker abel group and full Kostant-Toda lattice via micro-local anaysis. Guest investigates harmonic maps, quantum cohomorogy and mirror symmetry, and writes an introductory book Futaki proves the existence of Sasaki-Einstein metrics on some toric Sasakian manifolds, in particular, the existence of compelete Ricci-flat metric on the canonical bundles of toric Fano manifolds.

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Grant amount：\10400000 （ Direct Cost: \10400000 ）

We get the following results :
1.A maximal surface which is given by the real part of holomorphic isotropic immersion into C^3 is called a maxface. As a joint work with K.Yamada, the head investigator Umehara gave a Weierstrass-type representation formula for maxfaces, and gave an Osserman-type ineqality for complete maxfaces. The equality holds if and only if all ends of the surfaces are properly embedded. Moreover, as a joint work with K.Saji, S.Fujimori, and K.Yamada, the head investigator Umehara gave a criterion for the cuspidal cross cap, and showed that generic singular points for maxfaces consists of cuspidal edge, swallowtail and cuspidal cross cap.
2.As a joint work with K.Saji and K.Yamada, the head investigator Umehara studied the behavior of Gaussian curvature near the cuspidal edge and the swallowtail. In particular, the new geometric invariant on cuspidal edges called the singular curvature is introduced, and show that the integration of the singular curvature on the singular set is closely related to the Euler number of the surface.
3.A curve γ in the real projective plane is called anti-convex if for each point p on the curve, there exists a line passing through the point which does not meet y other than p. As a joint work with G.Thorbergsson, the head investigator Umehara studied the inflection points on anti-convex curves, and showed that the number of inflection points I and the number of the independent double tangents D satisfies the relation I-2D=3.

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Grant amount：\10800000 （ Direct Cost: \10800000 ）

The following results were obtained:
1) In a joint research project with U. Hertrich-Jeromin, S. Santos and F. Burstall, a suitable definition for discrete constant mean curvature surfaces in 3 dimensional space forms was obtained. Those 3 dimensional space forms consist of Euclidean 3-space, spherical 3-space and hyperbolic 3-space. It was shown that this new definition matches the old definition that is known for the Euclidean case, and this definition is new in the hyperbolic case. Using this definition, discrete Delaunay surfaces were studied, along with their discrete Darboux and Backlund transformations. An important tool in this research was the notion of conserved quantities. The case of smooth surfaces was developed by S. Santos and F. Burstall, while the discrete case was developed by U. Hertrich-Jeromin and myself.
2) In a joint research project with my Ph.D. graduate student N. Sultana, the stability and Morse index of constant mean curvature surfaces of revolution in spherical 3-space was studied. Because the axis of such a surface is a closed loop, these surfaces can become close tori, and then they will have finite index. It was shown that all such surfaces are unstable, and that they all have index at least 5, and (depending on the choice of surface) the index can be arbitrarily large. The index is the number of negative eigenvalues of the associated Jacobi operator.
3) In a continuation of a project with M. Kokubu, M. Umehara and K. Yamada, surfaces with constant Gauss curvature 0 in hyperbolic 3-space (flat fronts, which can have singularities) were studied. In particular, this year, it was shown that the caustics of such surfaces can have ends with asymptotic behavior described by cycloids.

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Grant amount：\3100000 （ Direct Cost: \3100000 ）

In this research, we planned to give a now development of the theories of classical differential geometry by restructuring them from the modern viewpoint, particularly, of the theories of integrable systems and of singularities. Our main results are the following :
1.(1)In affine differential geometry, one of the core theories of classical differential geometry, we mainly studied the geometry of affine hyperspheres and their representation formulae, and showed a relationship with the geometry of holomorphic statistical manifolds and the several properties of the center maps. We also studied the discretization of affine or centroaffine plane curves and gave a description of their time-evolution following discrete soliton equations ; (2)we characterized the classical examples of conformally flat hypersurfaces in 4-dimensional Euclidean space and constructed new examples ; (3)for real hypersurfaces in complex space forms, we introduced a new geometric invariant and classified Hopf real hypersurfaces using the invariant.
2.We studied the geometric properties of surfaces with singularities and obtained the following results : (1)We constructed the theory of flat fronts, the flat surfaces with singularities of a certain kind in 3-dimensional hyperbolic space. In particular, we defined (weak) completeness of flat fronts and showed their global properties ; (2)investigating the properties of the singularities of maximal surfaces in 3-dimensional Minkowski space, we constructed the theory of maxfaces, the spacelike maximal surfaces allowing singularities of a certain kind.
3.We studied transformations of surfaces and showed that the transformations given by the sphere congruences in Moebius geometry are obtained by the complexified line congruences in Euclidean space. We also investigated biharmonic curves in 3-dimensional homogeneous spaces and determined such curves when the homogeneous spaces are irreducible and reductive.

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Grant amount：\9200000 （ Direct Cost: \9200000 ）

1.W rewrote the Weierstrass-type representation formula for flat surfaces in hyperbolic 3-space in the form without integration (Darboux-type formula), and classified complete flat surfaces with small numbers of ends. 2.We pointed out the class of ambient spaces for which an analogue of Weierstrass-type (Bryant) representation formula for mean curvature one surfaces in hyperbolic 3-space holds. 3.We found criteria for singularities (cuspidal edges, swallowtails, cuspidal cross caps) which are generic singularities of fronts or frontals. 4.We established fundamental notions of flat fronts in hyperbolic 3-space, and investiagted properties of singularities of such surfaces. 5.We defined a certain class of maximal surfaces with singularities in Minkowski 3-space (called maxface), and investigated their singularities.

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Grant amount：\3900000 （ Direct Cost: \3900000 ）

We succeeded systematic constructions of families of anti-self dual (ASD) connections using representation theory of compact Lie groups before the project, which is a generalization of the ADHM-construction on the 4-dimensional sphere and Buchdahl's construction of instantons on the complex projective plane. Applying a method of dimensional reduction to our constructions, we can show that there is a relation between ASD connections on different base spaces. This method is expected to give a new way of finding vector bundles with ASD connections. It remains an important question whether our families of ASD connections are complete or not. This problem would be crucial in compactifying moduli spaces of ASD connections. We can succeed to construct a theory of twistor sections which is a section of a vector bundle satisfying the twistor equation. As a result, we obtain affirmative answers to the above question in various cases. This is because a twistor section corresponds to a holomorphic section on the twistor space, and we can apply homological algebraic methods to our problems. Moreover, when a theory of twistor sections is applied to homogeneous vector bundles on compact quaternion symnmetric spaces, we can show that there exists a bijection between the two sets. One is a set consists of zero loci of twistor sections and the others is the set of the real representations of simple compact connected Lie groups with non-trivial principal isotropy subgroups which are neither torn nor discrete groups. Using a theory of twistor sections, we can also show that there exists a relation between a singular ASD connection with a singular set and a vector bundle with such a connection. Here, a singular ASD connection naturally appears when we compactify the moduli spaces of ASD connections using the theory of monads. In short, we can show the fact in many cases that the homology class represented by the singular set of the singular ASD connection has a characteristic lass of a vector bundle as a Poincare dual. In higher dimensional cases, we necessarily meet the difficulty such that we need to consider too many sheaf cohomology groups on the twistor spaces when applying homological algebraic methods. Though we obtained vanishing theorems of sheaf cohomology groups before the project., we got more vanishing theorems which can be regarded as final versions. Combined these generalized vanishing theorems of sheaf cohomology groups with a theory of twistor sections, we can succeed to construct moduli spaces of ASD connections in more cases. Up to now, any systematic concrete examples of moduli spaces of higher dimensional instantons can not been seen anywhere except ours.

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Grant amount：\2800000 （ Direct Cost: \2800000 ）

I proved the homogeneity of isoparametric hypersurfaces with six principal curvatures with multiplicity two, which I had been tackling for several years. I also got a new proof of Dorfmeister-Neher's theorem which treats the multiplicity one case, in a unified manner.
Investigating the resulted homogeneous hypersurfaces, I got the following As was known in the case of multiplicity one, the hypersurfaces with 6 principal curvatures are given as a fibration over those with 3 principal curvature, where the fibers aret otally geodesic spheres. In the case of multiplicity two, the fiber dimension is six, while in the case of multiplicity one, this is three. Discovery of the fibration structure is an extension of our former results on the degenerate Gauss mapping which was done with G. Ishikawa and M. Kimura.
Moreover, using the fact that the family of isoparametric hypersurfaces fill the ambient space, we get an interesting relation between 13-dimensional sphere and 7-dimensional sphere. Furthermore, using that these hypersurfaces are given as orbits of the exceptional group G_2, we can show that there exists a metric on S^7-CP^2 of which holonomy group is G_2. From this, a real open version of Calabi conjecture will be considered, i.e., when a compact Riemannian manifolds with positive Ricci curvature from which a certain part removed, admits a metric with G_2 holonomy? In this way, hypersurfaces obtained as G_2 orbits suggest us very important and interesting problems.

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Grant amount：\9600000 （ Direct Cost: \9600000 ）

1.We have completed the study of collapsing 4-manifolds whose sectional curvature and diameter are uniformly bounded from below and above respectively, and established the geometry of 3-dimensional and 4-dimensional complete open spaces of nonnegative curvature (Yamaguchi).
2.We have proved that a 3-manifold with a lower curvature bound having a small volume is a graph manifold (Yamaguchi and Shioya).
3.We have determined the Gromov-Hausdorff convergence of surfaces with uniformly bounded total absolute curvature, and developed geometry of limit pearl spaces in detail such as singularities, homotopy types, number of pearls (Yamaguchi and Hori).
4.We have determined local geometric properties of a neighborhood of a singular point in an two-dimensional singular spaces with curvature bounded above proving that it is a gluing of several Lipschitz disks (Yamaguchi, Nagano and Shioya).
5.We have defined the notion of singular spaces with Ricci curvature bounded below, and introduced energy forms from such spaces to general metric spaces. We have proved the Poincare inequality and a compactness theorem using it (Kuwae and Shioya).
6.We have considered discrete approximations of spaces like Riemannian manifolds or Alexandrov spaces by graphs called nets, and proved that the convergence of Laplacians of nets to that of the space (Otsu). Using the idea of net-approximation above, we have studied the asymptotic behavior of heat operators on manifolds and obtained a central limit theorem for heat operator on nilpotent covering manifolds.

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Grant amount：\4000000 （ Direct Cost: \4000000 ）

We get the following results:
1. The head investigator Masaaki Umehara proved that the equality of the Osserman inequality for minimal surfaces in Euclidean n-space holds if and only if each end has no self-intersection and asymptotic to a catenoid or a plane, which is a joint work with M. Kokubu and K. Yamada. Moreover, we construct a new example which attains the equality.
2. The head investigator Masaaki Umehara gave countable many new irreducible constant mean curvature one (i.e. CMC-1) surfaces in hyperbolic 3-space whose ends all irregular, which is a joint work with W. Rossman and K. Yamada.
3. The head investigator Masaaki Umehara gave an elementary proof of the Small's representation formula for CMC-1 surfaces in hyperbolic 3-space and also got a similar representation formula for flat surfaces in hyperbolic 3-space, which is a joint work M. Kokubu and K. Yamada. A flat surface in hyperbolic 3-space called a flat front if it admits singularity but can be lifted to a Legendrian immersion into the unit cotangent bundle of hyperbolic 3-space. We showed flat fronts with complete ends satisfy an Osserman-type inequality with respect to the degree of the hyperbolic Gauss maps. The equality holds if and only if all ends have no self intersection. Furthermore, we classify flat front of genus zero with embedded regular 3-ends and also construct an example of genus one with five regular embedded ends.

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Grant amount：\3700000 （ Direct Cost: \3700000 ）

We tried to investigate the following themes or questions with concernig of the finite type submanifolds or biharmonic submanifolds :
(i)Are there the finite type surfaces with given mean curvature H in the family of surfaces of revolution x.(u, v)=(u cos v, u sin v, f(u)) generated by the periodic function z=f(u) ?
(ii)Are there the finite type surfaces with given Gauss curvature K in the family of surfaces of revolution x(u, v)=(u cos v, u sin v, f(u)) generated by the function z=f(u) ?
(iii)Are there the finite type surfaces with constant mean curvature in the family of surfaces ?
(iv)Are there the Willmore surfaces of finite type ?
(v-1)The finite type submanifolds in the space with D` Atri metric.
(v-2)The biharmonic submanifolds in the space with D` Atri metric
We will continue to study in future the following still open conjectures of Prof. Bang-yen Chen
1.To determine all of the finite type surfaces in Eucldean space of dimension 3
(Chen conjectur 1 : The only finite type surfaces in Eucldean space of dimension 3 are the minimal suefaces, spheres and right cylinders.)
2.To determine all of the biharmonic submanifolds in Eucldean space of dimension n
(Chen conjectur 2 : The only in Eucldean space of dimension n (n>3) are the harmonc ones.)
3.To determine all of the biharmonic submanifolds in Minkowsky space of dimension 4.

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Grant amount：\1900000 （ Direct Cost: \1900000 ）

In this research, we studied classical differential geometries, theory of integral systems and information geometry.
1. Classical Differential Geometries (1) We characterized minimal affine hypersurfaces and minimal centroaffine immersions of codimension two. Moreover, we gave an explicit method of constructing self-dual minimal centroaffine surfaces of codimension two.
(2) We studied manifolds with projectively flat torsion-free affine connection whose Ricci curvature is symmetric and definite, and showed fundamental results on the injectivity of the projective developing maps of such manifolds and the convexity of their image.
(3) For conformally flat hypersurfaces of a 4-dimensional sphere, we defined a new conformal invariant. Using the invariant, we characterized the classical examples and constructed new examples.
(4) We developed a very concrete and comprehensive theory on curves and surfaces in 3-dimensional homogeneous spaces.
2. Integrable Systems We investigated various integrable systems appeared in classical differential geometries. We obtained representation formulae for minimal surfaces in 3-dimensional solvable Lie groups and flat surfaces in a 3-dimensional hyperbolic space. We also developed a comprehensive theory of (spacelike) surfaces with harmonic inverse mean curvature in 3-dimensional Riemannian space forms and Lorentzian space forms.
3. Information Geometry and Statistical Manifolds (1) We defined complex statistical manifolds and studied them from the view points of affine differential geometry and of information geometry, especially of quantum estimation theory.
(2) As a generalization of special Kahler manifolds, we defined statistical manifolds with compatible complex structure and investigated their fundamental properties.
(3) On (-1)-conformally flat statistical manifolds, we gave an explicit method of constructing the Volonoi diagrams.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

(1) Working with Konrad Polthier, we considered a variational approach for defining discrete minimal surfaces, and established a variational approach for defining discrete constant mean curvature surfaces. We constructed discrete catenoids and helicoids and Delaunay surfaces, and we completely classified the case of catenoids. Furthermore, we computed the discrete Morse index of discrete minimal surfaces and used these computations to examine the Morse index of smooth minimal surfaces. (2) The head investigator proved that Wente tori (these are genus 1 compact constant mean curvature surfaces in R^3) have Morse index at least 7, and also found a lower bound for the Morse index of Wente tori that grows with the spectral genus of the surface. Furthermore, working with Lima and Sousa Neto, we improved the lower bound estimate of 7 to 8. (3) Working with Lima and Berard, we determined the growth rate of the Morse index on complete noncompact constant mean curvature surfaces. (4) Working with Thayer and Wohlgemuth, we constructed many examples of doubly-periodic minimal surfaces in R^3. (5) Working with Umehara and Yamada, we classified all constant mean curvature 1 surfaces in hyperbolic 3-space H^3 that have total curvature at most 8π. (6) Working with Umehara and Yamada and Kokubu, we have started a project to study the nature of singular points on flat surfaces in H^3. (7) Working with Schmitt and Kilian, we have started a project to study constant mean curvature surfaces of genus 0 with three asymptotically Delaunay ends in R^3 and H^3 and the 3-dimensional spherical space S^3.

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Grant amount：\12500000 （ Direct Cost: \12500000 ）

The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometric functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.
For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function.

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Grant amount：\3600000 （ Direct Cost: \3600000 ）

Let us denote by A the space of Alexnadrov spaces of bounded curvature below and Hausdorff dimension above equipped with Hausdorff distance and by I the space of upper-semicontinuous functions on A. We call I the space of invariants. An ordered finite set of points of metric space is called a net, which is a discretization of the metric space. Since the configuration space of all nets is identified with the product of the space, the set N of pairs of spaces in A and its nets can be interpreted as a fiber space over A. We consider a map that assign the matrix of mutual distances of two points for each net. In this way we can represent N as a subspace of some Banach space. Then we introduce other maps form N to some Euclidian space that take local information of the above distance matrix. Especially we defined discrete Laplacian similar to the Laplacian of functions of Riemannian manifold. We introduced new statistical method to take average of discrete Laplacian on configuration space of nets. In this way we have showd that the eigenvalues and eigenvectors of discrete Laplacian converge to the limit independent of the choice of nets ; we also proved that coincides with the Laplacian in the sense of Kuwae-Machigashira-shioya in some sense.
Next we defined new structure on A by comparing two discrete Laplacian of different spaces and nets because they are same member of matrix space. Since in information geometry the relative entropy of two distributions determines Reimannian metric, we first introduced stationary Markov chain form the Laplacian, then we apply the relative entropy for them; finally we construct continuum limit of them, which is a generalization of Hausdorff distance.

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Grant amount：\3100000 （ Direct Cost: \3100000 ）

1. The Morse Index of Wente Tori
We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planer principal curves. We then prove a result that gives an algorithm for computing the index sharply.
2. Doubly Perodic Minimal Surfaces
We consider the question of existence of embedded doubly periodic minimal surfaces with Scherk-type ends. We extend the existance results of Karcher and Wei to more cases and we find other cases where existance dose not hold.
3. Embeddedness of area-minimizing disks
We show that a polygonal Jordan cave C satisfies certain conditions, then the least-area Douglas-Rado disk with boundary C is unique and is a smooth graph. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces.

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Grant amount：\15100000 （ Direct Cost: \15100000 ）

In this project, we had several sizes of symposium, international conferences, workshops etc. 3 times in 1999, 3 times in 2000, 2 times in 2001. Especially, we organized the 46-th Geometry Symposium at Univ. of Tokyo in August, 1999, the 9-th MSJ-IRI " Integrable Systems in Differential Geometry " in July, 2000, and an international workshop on Geometry of Riemannian Submanifolds at Tokyo Metropolitan Univ. in December, 2001.
There we were able to have talks of results, discussions and interchanges of informations about differential geometric research on submanifolds synthetically and to receive the review for this project. Ogiue, Maeda, Adachi developed study of curves, in particular circles, in more general symmetric spaces and their submanifolds than complex space forms, and obtain the results. Especially their research examining in details the length spectrum of geodesic spheres in complex space froms by the number theoretic method were highly estimated and published in J. Math. Soc. Japan. Miyaoka gave a different proof of homogeneity for the case six principal curvatures and multiplicity 1 in the classification problem of isoparametric hypersurafces and her work got the high estimate. Further in cooperation with Ishikawa, and Kimura, she constructed many compact submanifolds in spheres with degenerate Gauss maps satisfying the equality of Ferus and gave many examples of special Lagrangian submanifolds as a by-product. Kitagawa studied isometric deformations of flat tori in 3-dimensional standard sphere with nonconstant mean curvature and proved that these flat tori are all deformable. Kenmotsu classified parallel mean curvature surfaces with constant Gaussian curvature in complex space forms by analyzing precisely a certain nonlinear ordinary differential equation and his work was decided to be published in an internationally high journal. In joint works with Hashimoto and Mashimo, Sekigawa gave the structure equations for 4-dimensional CR-submanifolds in the nearly K\"ahler 6-dimensional sphere by the moving frame method in Bryant's style, and as applications he studied fundamental properties of characteristic classes of such submanifolds and provided new examples by the Lie theoretic method. Koiso analyzed actively deformation and stability problem of constant mean curvature surfaces with boundary and obtained excellent results. On the other hand, Ogiue, Nakamula, Hamada greatly have maken good use of the system (PPDG) interchanging extensively and effectively research results and informations, and we expect that this system will be developed to the Geometry Server in future by Guest and Ohnita.

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Grant amount：\2100000 （ Direct Cost: \2100000 ）

今年度の研究の結果,次の点が明らかとなった.
・曲率が下に有界な3次元コンパクト・アレクサンドロフ空間において極小曲面を構成するためには,そのような空間にリプシッツ構造を構成する必要があった.これに関して,一点からの距離関数の一般化された意味の積分曲線のリプシッツ性が得られた.とくに,2次元の場合には距離球にリプシッツ構造が入るので,アレクサンドロフ曲面にはリプシッツ構造が入ることが分かった.3次元において問題となるのは,距離球にリプシッツ構造を構成する点である.これは2次元の場合の結果を拡張することで実現出来そうである.従って,これまでの研究の結果により,3次元アレクサンドロフ空間にリプシッツ構造を構成する問題は,かなり現実的に解決可能な問題となってきた.
・上記問題と少し関連して,絶対全曲率が押さえられた曲面が,測度つきグロモフハウスドルフ収束に関する収束,崩壊の具体的な記述が可能となった(堀敦彦氏との共同研究).来年度中に論文を完成させる予定である.
・最大の頂点数をもつコンパクト非負曲率アレクサンドロフ空間の等長類の分類については,数学的には出来ているのだが,本年度は論文を完成する時間的余裕がなかった.来年度中に完成させる予定である.

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Grant amount：\5900000 （ Direct Cost: \5900000 ）

We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space.
As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π.
On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature.

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Grant amount：\6600000 （ Direct Cost: \6600000 ）

1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.
2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the conformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.
3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.
4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

We obtained 25 and more prints in this Project (See REFERENCES).
In 1997
1. We obtained some new results about the classification of biharmonic submanifold in psudo-Euclidean space. In detail,
(1) The classification problem of biharmonic curves in psudo-Euclidean space was completed.
(2) It was proved that the bihaemonic surfaces do not exist in 3 dimensional psudo-Euclidean space.
(3) Some classification theorems of biharmonic surfaces in 4 dimensional psudo-Euclidean space was obtained.
2. About the spacelike maximal submanifolds with some conditions for Ricci curvature immersed in de-Sitter sphere in psudo-Euclidean space, the classification of them was discussed.
3. About the hypersurfaces of constant scalar curvature immersed in de-Sitter sphere in psudo-Euclidean space, the sphere theorem was discussed.
4. About the comformally flat 3 dimensional Riemannian manifolds under some conditions for Ricci curvature and scalar curvature, the classification problem of them was discussed.
In 1998
1. We obtained, under some conditions for scalar curvature, that any compact submanifold immersed in de-Sitter sphere in psudo-Euclidean space is only a standard sphere.
2. We obtained a characterization about the Clifford torus.
3. (1) We discussed about 3-dimensional comformally flat Riemannian manifold with non negative constant scalar curvature and the constan norm of Ricci curvature.
(2) We discussed about 3-dimensional comformally flat Riemnnian manifold with negative constant scalar curvature and the constan norm of Ricci curvature.
In 1999
1. We discussed the classification problem about the minimal closed surfaces in unit sphere with bounded norm of Ricci curvature. This result is concerted with the famous theorem by S.S.Chern, do Carmo and S. Kobayashi that the Clifford torus is only minimal closed surfaces of S=n in unit sphere.
2. We obtained some progress concerned with the third work of listed in 1998
3. We now investigate the following open problems proposed by Bang-yen Chen;
(1) The classification problem of the finite type surface in 3 Euclidean space.
(2) The classification problem of the biharmonic submanifolds in n-dimensional Euclidean space.
(3) The classification problem of the biharmonic submanifolds in 4-dimensional Minkovski space.

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Grant amount：\3700000 （ Direct Cost: \3700000 ）

The objecitve of this project is to study the general hypergeometric functions (GHF) which were introduced by us to give a unified understanding of the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function and to give a natural generalization to the case of several variables.
1 : GHFs are defined as solutions of certain holonomic systems on the Grassmannian Gr_<r, n> and they have the integral representations in a formal sense whose integrand is a multivalued function on P^r. To obtain explicit resutis on GHF, it is important to understand this integral representation in the framework of de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here, for the integral on P^r, we defined the homology group as a locally finite homology group and then show that it is isomorphic to the relative homology group with compact supports for some pair of subsets P^r. Moreover, using this result, we computed explicitly, in the case r=1, the dimension of the homology group and gave a basis of the group.
2 : For the Beta function B(alpha, beta), the simplest case of GHF with regular singularity, and for the Gamma function GAMMA(alpha), the simplest case of GHF with irregular singularity, the following formulas are well known :
B(alpha, beta)B(-alpha, -beta)=2pii(<@D71(/)alpha@>D7+<@D71(/)beta@>D7)(<@D7-e<@D12pii(alpha+beta)@>D1-1(/)e<@D12piialpha@>D1-1(e<@D12piibeta@>D1-1)@>D7), gamma(alpha)gamma(1-a)=<@D7pi(/)sinpialpha@>D7
We investigate the problem of understanding the above formulas from the viewpoint of de Rham theory. Explicitly we try to understand the right hand sides of the above formulas as a product of cohomological intersection number and the homological intersection number. For the GHF defined by the 1-dimensional integral, we computed explicitly the intersection matrix for the cohomoloy group by choosing its good basis.
By the choice of good basis, we can show that the intersection matrix turns out to be independent of the variables of the general hypergeometric function. The main reason for the computability of the intersection numbers is that the good basis has, at each singular point of the connection form of the de Rham complex, the analogous properties to the flat basis of the Jacobi ring for the simple singlarity of A-type.

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Grant amount：\3400000 （ Direct Cost: \3400000 ）

We investigated a global properties of Weierstrass representation. First of all, we studied a fundamental problem related to global problems. In particular, as the monodromy problem for minimal surfaces in Euclidean geometry is considered as a period problem of certain integral of holomorphic forms, that of CMC-1 surfaces in hyperbolic space can be considerd as a monodromy problem of a ordinary differential equation on Riemann surfaces. In this context, the monodromy problem is the condition for SL(2, C)-monodromy group to be reduced to the unitary group. To find a criterion of such a condition is difficult. However, when a problem satisfies some symmetric properties, it can be solved explicitly. Using this fact, we have constructed a large amount of examples of CMC-1 surfaces. Related to this problem, we investigated metrics of constant positive curvature with conical singularities, and obtained a classification result.
Related to classification of CMC-1 surfaces, we defined a homology invariant on CMC-1 surface, which is called flux, and proved some non-existence theorem using this invariant. Moreover, using Weierstrass representation for maximal surfaces in Minkowski 3-space, we have classified of surfaces with cone-like singularities with some finiteness.

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本研究では、一般合流型超幾何関数全体のなすヒエラルキーの構造を、様々な視点から解明していくことを目的としていた。今年度得られた成果は次の通りである。
1.合流型超幾何関数に付随する(コ)ホモロジー群の交点理論について。
確定型の超幾何関数に対しては、(コ)ホモロジー群の交点理論は確立されつつあり、交点数を具体的に計算する手段も知られている。合流型に対しては、一つは確定型の場合の方法で類似することで、交点理論が建設されつつある。一方、交点理論からの帰結として(確定型)超幾何関数の2次関係式が得られていたが、その関係式に直接合流操作を施すことで、合流型の超幾何関数についての2次関係式を手に入れることができた。その結果から、逆に交点理論のあるべき姿を予測することができるようになった。
2.局所・大域解析について。
一般合流型超幾何関数は、初等関数を核とする積分表示を持つので、積分サイクルを決めたときの局所挙動を調べることができれば、局所解析と大域解析が結びつくことになる。(2、4)型、(2、5)型の合流型超幾何関数について、サイクルの合流を追跡することで、典型的な挙動を与えるサイクルは、合流後もやはり典型的な挙動を与えるサイクルになること、その背後には漸近挙動の合流があることを解明した。
3.合流操作の応用
線形微分方程式のモノドロミ-群に対して、線形アーベル群の合流を応用し、退化した場合の結果を退化していない場合の結果から極限操作で得ることができた。

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Grant amount：\2900000 （ Direct Cost: \2900000 ）

The objective of this project is to study the general hypergeometric functions (GHF) which were introduced by us to treat the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function.
1 : GHFs are defined as solutions of certain holonomic systems on the Grassmannian Gr_<r, n> and they have the integral representations in a formal sense. We try to understand these integrals in the framework of the de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here we treat this problem in the particular cases of GHF,the case of generalized Airy functions and the case of GHFs given by the one dimensional integrals.
(1)In relation to the problem of expressing the holonomic system for the generalized Airy functions as the integrable holonomic connection outside of the singlar locus, we computed in [3] the cohomology group of the rational twisted de Rham complex associated with the representation. We showed that the cohomology groups vanish except for the r-th one and that dim H^r=_<n-2>C_<r-1>. Moreover we presented the conjecture that a basis of H^r is given in thems of the Schur functions.
(2)We understand domains of integrations for the generalized Airy integrals as cycles of a homology group on P^r with the family of supports defined by the integrand. By using the r-dimensional saddle point method, we showed in [4] that the homology groups are trivial except for the r-th one and that r-th homology group forms a local system of Z-modules on the space of independent variables of the functions rank _<n-2>C_<r-1>.
(3)In the case where the GHFs are given by the one dimensional integrals (in other terms, the confluent case of Lauricella's F_D), we showed that the rational de Rham cohomology groups are trival except for H^1, and gave a basis of H^1 explicitly.
2 : It is known that the other special functions of confluent type are derived from the Gauss hypergeometric function by the limit processes called confluences. In [5] we showed that this phenomenon can be explained by the adjacency relations among the strata of the stratification naturally introduced in the set of regular elements in the Lie algebra gl_n. Furthermore we generalized the above limit process to GHF in general.

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Grant amount：\8300000 （ Direct Cost: \8300000 ）

Kenmotsu has published a paper, in which he proved theorems for intersections of minimal submanifolds in manifolds with partially positive curvature. Kenmotsu is studying local behavior of the Kaehler angles of minimal surfaces with constant Gaussian curvature in two dimensional complex space forms : In order to classify such minimal surfaces, at first we obtained differential geometric characterization of the second fundamental forms of such minimal surfaces. By using it we obtained an overdetermined system for the Kaehler angle. This is reduced to a system of two ODE's. By the values of the Gaussian curvature of the surface and the curvature of ambiant space, these systems are different. We developed analysis to these systems extensively and proved that they have no non trivial common solution even locally. It implies local classification theorem of suchminimal surfaces.
Fukaya has proved the Arnold conjecture in the general setting. This is really exciting.
For reserch of submanifold geometry, R.Miyaoka has studied relations between minimal surfaces in complex projective spaces and the Toda equations extensively and published her results in the Crelle Journal. Yamada has contributed to construct the theory of constant mean curvature surfaces in the hyperbolic spaces.
For the reserch of global Riemannan geometry, Suyama has given a new method to construct diffeotopy of standard spheres and applying it he proved a differentiable pinching theorem for 0.654 pinched compact riemannan manioflds. T.Sakai has written a textbook of global Riemannian geometry which was published by the American Math.Society.

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Grant amount：\1200000 （ Direct Cost: \1200000 ）

双曲型空間の定平均曲率1の曲面(CMC-1曲面)に対して「双対性」という概念が発見された。この概念を用いることにより完備・有限全曲率をもつようなCMC-1曲面で複雑な位相型をもつ(種数の高い)ものが数多く構成できた。これらの曲面に付随して,種数の高いコンパクトリーマン面上の錐的特異点をもつ定曲率1の共形計量の例を作ることができた。
これらの曲面,計量及びCMC-1曲面のWeierstrass表現の変形を通して双曲型空間の定平均曲率曲面が得られるだろう。双対性の概念と表現公式を通してその変形の方向に対するひとつの候補をみつけることができた。
双対性の概念を用いると,Gauss写像がCMC-1曲面の性質にどのような影響をあたえるか,が具体的にわかる。このことを用いて「全曲率」に関する不等式を得ることができた。

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Grant amount：\900000 （ Direct Cost: \900000 ）

双曲型空間H^3の中の定平均曲率1の曲面(CMC-1曲面)について,正則なデータを用いてそれを表現するBryantの公式-Weierstrass表現の双曲版-が知られていたが,それから具体的に曲面を構成することは難しい.それは,その表現公式にデータとして与えられる正則関数の幾何学的意味がimplicitであることに起因する.そこで,幾何学的な意味が明白な量であるような量で,双曲的Gauss写像,Hopf微分を用いて曲面を表現することを考えた.実際,これらの量を与えられると,対応するCMC-1曲面全体の集合を決定することができる.さらに,それらの曲面は,(簡単のため完備,全曲率有限の場合に限れば)コンパクトリーマン面上の定曲率1の共形擬計量で,有限個の点に錐的特異点を持つものと1対1の対応が付くことがわかった.この対応によって,完備,全曲率有限なCMC-1曲面で,対称性の高いものが構成できた.これらは,Euclid空間における対称性の高い極小曲面-例えばJorge-Meeks曲面-の双曲的対応物であるが,極小曲面の場合と違い,自己交差を持たないことが有り得る,ということが,数値計算とコンピュータ・グラフィックスによる実験の結果予想できた.これは,双曲型空間におけるCMC-1曲面の,Euclid空間の極小曲面と大きく異なる性質であると考えられる.
われわれの構成法における双曲的Gauss写像の役割を観察すると,平均曲率が1でない場合の表現公式に関する示唆が得られる.しかし,この計算を具体的に遂行するには今一つ時間がかかると考える.

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Grant amount：\1000000 （ Direct Cost: \1000000 ）

3次元双曲型空間の平均曲率1をもつ曲面について、Bryautの表現公式を見直すことによって、曲面と、Conical singularitiesをもつ定曲率1をもつRiewann面上のConfonnal metricとの関係が明らかになった。
このことと関連して、irreducibleな曲面-いままでひとつも例がしられていなかった-の例を構成することができた。これらの例はEuclid空間の極小曲面の中に対応するものが知られているが、数値計算とコンピュータグラフィクスの結果から双曲型空間の例の中にEuclid空間の場合とことなり、embeddedになるものが存在することが予想できた。このembeddednessの液学的証明は今後の課題である。
また、最近とくに多く知られるようになったEuclid空間の極小曲面を、“Small perturbatim"-空間型の変形とLie群の変形理論による-によって双曲型空間の平均曲率1の曲面に変形する一般的な方法への糸口がみつかった。いくつかの具体例についてはこの方法は成功している。

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Grant amount：\1400000 （ Direct Cost: \1400000 ）

標記研究課題の今年度の経過及び成果を次のように報告し、今後の展望を簡単に記す。
交付申請書においては、研究の目的が大きく分けて4個あった。そのうちの一つに、「ブラウン運動の正値超汎関数が決める一次形式は有界正値測度による積分形式となるが、この定理をより精密化することによって、ブラウン運動の正値超汎関数に対応する正値測度をFernique型の評価式で必要十分条件として特徴付ける。」を挙げておいたが、この問題に関してほぼ目的通りの成果を得る事が出来た。詳しく言うと、少し一般化して、ホワイトノイズの正値超汎関数(Hida Distribution)に対して、それに対応する正値測度をFerniqu型の評価式で特徴付けることが出来た。なお、この成果を得るに当たって、補助金からの旅費による出張がもたらした研究上の情報交換が重要であった。特に、昨夏の名古屋でのConference on Gaussian Random Fieldsにおいて、Potthoff,Streit(Bielefeld,独)から与えられたBerezanskii,Kondrat'ev,Samoilenko(Kiev,ソ連)等の研究に関する情報は貴重であった。枠組み、方法、視点は、異なるが、多くの結論が独立に得られていることが分かった。また、この研究会議中に、Berezanskii氏と直接議論をすることが出来、上記成果の発表に関して、氏からの示唆を受けた。この成果については昨年10月にある雑誌に投稿し、現在referee,editor、著者との間で連絡調整中である。
この課題研究の大きな目標の一つであるFeynmanの経路積分に関しては、Donsker's delta functionが正値超汎関数になるので、それに対応する測度による積分を具体例に対して、試行錯誤的に適用し、種々の考察をしようとしているところである。

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Grant type：Competitive

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Grant type：Competitive

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