(1)We investigated the convergence and collapsing phenomena of Riemannian manifolds with boundary whose sectional curvatures are uniformly bounded below and the second fundamental forms of the boundaries are unifowmly bounded. We characterized the boundary singular points of the limit spaces, and determined the geometry of the limit spaces. (2)We determined the local structure of geodesically complete two-dimensional metric spaces with curvature bounded above, based on the topological singular point set. We also obtained an approximation theorem by polyhedral surfaces and the Gauss-Bonnet theorem.(3) We determined the topology of three-dimensional Alexandrov spaces with boundary. <BR>

During this project, Fujiwara with his joint work with Bestvina-Bromberg introduced the theory of Projection complex, and found several important applications. For example, we proved that a mapping class group acts on a finite product of quasi-tree with a QI-embedding orbit, and as a consequence it has finite asymptotic dimension. By now the technique of projection complex became an important tool in Geometric group theory. Ozawa in his joint work with Kaluba-Novak proved that the automorphism group of the free group of rank 5 has property T, using computer. This settles a long standing problem. It also opens a new direction of research.

Based on the study of concentration of measure phenomenon, Gromov proposed a new geometric theory for metric measure spaces. One of main motivation to study this theory is to investigate a sequence of spaces with unbounded dimension. In our study, we develop and deepen the theory. The concentration of measure phenomenon can be considered as a geometric variant of the law of large numbers. We study a geometric variant of the central limit theorem, which appears as an analog of phese transition phenomenon. One of examples is the sequence of spheres with unbounded dimension. If the radius of the sphere has small order, then we observe the concentration of measure phenomenon. If the radius has large order, then we observe the dissipation phenomenon. If the radius has the order of the square root of the dimension, then we see that the sphere converges to the infinite-dimensional space with Gaussian measure. We have proved this kind of phenomenon for many other spaces.

1. We properly defined the notion of good coverings of Alexandrov spaces, and obtained a Lipschitz homotopy convergence theorem in the non-collapsing case using it. 2. In the case when a manifold with boundary inradius collapses under a lower sectional curvature bound and a two-side bounds on the second fundamental form of the boundary, we determined the manifold structure. This gives an extension of a result due to Gromov, Alexander-Bishop. We also determined the structure of inradius collapse of codimension one in the case of bounded diameter. 3．We developed geometric analysis of metric measure spaces concerning isometric inequalities and spectral inverse problems about lattices and surfaces of revolusion.

We study mathematical framework for Quantum spin system. Physics on topological Insulator and topologically protected surface/edge state is formulated in K-theory. By using the non-commutative geometry, we generalized it to disordered systems. We also develop discrete surface theory to study the relation of microscopic structural data and macroscopic properties and apply it to carbon networks.

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.

Fujiwara found a new method to embed certain discrete groups into a finite product of hyperbolic graphs in his joint work with Bestvina and Bromberg, and obtained important applications. Ozawa made a significant progress on the study of C*-algebra of discrete amenable groups, and also quasi-homomorphisms into non-commutative groups. Shioya found an important result on the asymptotics of the first eigen value of Laplacians on Alexandrov spaces.

We obtained a Jensen's inequality over complete p-uniformly convex spaces. Further we proved the unique existence of the (non-linear) resolvent associated to a coercive proper lower semi continuous function satisfying a weaker notion of p-uniform convexity on a complete metric space and establish the existence of the minimizer of such functions as the large time limit of the non-linear resolvents, which generalizes the pioneering work by J. Jost for harmonic maps into CAT(0)-spaces. The results can be also applied to Lp-Wasserstein space over complete separable p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces. On the other hand, we investigated the spectral bounds for symmetric Markov chains with positive n-step coarse Ricci curvature for not only functions but also maps into complete separable 2-uniformly convex spaces with geometric conditions.

We obtained the uniqueness and the stability of the inverse spectral problem concerning the local data of the heat kernels in the moduli of closed Rimennian manifolds whose sectional curvature and diameters are uniformly bounded (T.Yamaguchi, S. Kurylev, M.Lassas). We classified the collapsing phenomena of three-dimensional closed Alexandrov spaces with cuvature uniformly bounded below. Moreover we proved the local strong Lipschitz contractibility of Alexandrov spaces and the stability of strongly Lipschitz contractible balls(T.Yamaguchi, A.Mitsuishi). We proved that the curvature dimension condition of Ricci curvature is preserved under the concentration of metric measure spaces(T.Shioya, K.Funano). We reconstructed hyperbolic orbifolds from S-matirix corresponding to a general end (H. Isozaki, Y. Kurylev).

We studied the details of a geometric theory of metric measure spaces due to Gromov and wrote a book for it. We proved that if a sequence of metric measure spaces converges to a metric measure space with respect to the observable distance, then the curvature-dimension condition is stable. As an application, we give an estimate of the ratio of the k-th eigenvalue and the first eigenvalue of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature, where the estimate depends only on k. Gromov defined a natural compactification of the space of metric measure spaces with the observable distance. We deeply considered it and introduce a new metric structure on it. We apply our metric structure to prove that an n-dimensional sphere of radius square root of n in a Euclidean space converges to an infinite-dimensional Gaussian space as n tends to infinity.

We studied properties of eigenavalues of Laplacian on closed Riemannian manifolds of nonnegative Ricci curvature. One of our achievements is the k-th non-trivial eigenvalue of Laplacian on such manifolds is bounded by the first eigenvalue times universal constant depending only on k. In our proof, we obtained a stability result of curvature-dimension condition under concentration topology. This result extends the known-result that cuvature-dimension condition is stable under the measured Gromov-Hausdorff topology.

Aim of this research proposal is to develop new methods to study geometric objects with singularities, or discrete spaces, which are not accessible by traditional differential geometrical technics. Our idea is to apply probability theory to those geometric objects. Some results are obtained and published from international journals.

We prove that given an Alexandrov space and a positive Radon measure on it, if the measure satisfies a comparison condition of Bishop-Gromov type and if the space contains a straight line, then the space is homeomorphic to the direct product of some space and the real line. This is a generalization of the Cheeger-Gromall splitting theorem. As another result, given a sequence of closed Riemannian manifolds of nonnegative Ricci curvature and with a uniform upper bound of diameter, if the k-th eigenvalue of the Laplacian of the manifold in the sequence is divergent to infinity, then the first eigenvalue is also divergent and the measure concentration happens.

Hyperbolic group was invented by Gromov in 80's. We aim to use hyperbolicity in broader objects. We constructed many quasi-homomorphisms on Kac-Moody groups. We obtained a first finitely presented, simple group with infinite commutator width (with Caprace). Using quasi-homomorphisms, we characterized rank-1 manifolds among complete Riemann manifold of non-positive curvature of finite volume (with Bestvina).

In view of the theory of Markov processes and Dirichlet forms, we investigate geometric singular spaces. In particular, analysis on harmonic maps and (sub-) harmonic functions is the main aim of this research. Some of our results extend the previous results of the research supported by Grant-in-Aid for Scientific Research (C) No.16540201 from Japan Society for the Promotion of Science. We also obtain some results on the theory of Markov processes as a byproduct.

In these days, the study of geometric analysis on metric measure spaces is going around. The head investigator, Shioya, Studies such a subject and his main interest is curvature of metric measure spaces and convergence, especially Alexandrov spaces, Ricci curvature of metric measure spaces, and Gromov-Hausdorff convergence of metric measure spaces. On he other hand, Mosco studied variational convergences, which is a functional analytic theory of convergence of Dirichlet energy forms. We, Shioya and the investigator, Kuwae, thought that Mosco's theory is deeply related with the study of convergence of metric measure spaces, and have extended the theory in the geometric viewpoint. We have completed it in the period of this project. The concept of convergence in our theory is nowadays called the Mosco-Kuwae-Shioya convergence and is being widely applied to the finite dimensional method in probability theory and also to some homogenization problems. Another study is on a Laplacian comparison theorem and a splitting theorem on Alexandrov spaces with some condition corresponding to a lower bound of Ricci curvature. This is still on going. For Riemannian manifods, the Ricci curvature being bounded below is equivalent to an infinitesimal version of the Bishop-Gromov inequality. Since it is impossible to define the Ricci curvature tensor on Alexandrov spaces, we consider the infinitesimal Bishop-Gromov inequality instead of the Ricci curvature bound. Different from Riemannian, the cut-locus is not necessarily a closed set in an Alexandrov space. That may even be a dense set. By this reason, the same proof as for Riemannian manifolds does not work and we develop a new method of proof.

We discuss ed a long time behavior of periodic random walks on a crystal lattice in view of geometry, a large deviation property in particular, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as remarkable combinatorial features,. A crystal lattice has a metric structure with the graph distance. By changing scale of the distance, we obtain a one-parameter family of metric spaces. The Gromov-Hausdorff limit of the sequence is called the asymptotic cone at the infinity of the crystal lattice. As the scale go to zero., because of the periodicity of the crystal lattice, the asymptotic cone exists and we determinded its unit ball explicitely in terms of combinatorial data. We also published a survey article on discrete geometric analysis of crystal lattice from Sugaku Expository, Amer.Math.Soc. In there, we discussed spectral properties and geometry of random walks on a crystal lattice, such as the law of large number, the central limit theorem, large deviation and spectrum of magnetic Schroedinger operators from non-commutative geometry.

1. We gave a necessary and sufficient condition for stochastic processes in a class of one-dimensional Markovian (not necessarily strong Markovian) processes with continuous paths to be bi-generalized diffusion process which were introduced by the head investigator. It is worth to note that the scale functions of the processes in our class are no more continuous in general and may miss the strong Markov property at the discontinuous points of scales. 2. We gave large deviation principles for sums of independent identically distributed fuzzy set-valued random variables with compact level sets. More precisely, we gave Cramer type large deviation principles for such sums with respect to the topologies induced by the mean convergence of order p, Levy's metric and graph convergence, and the relative topology of a countable direct product topology. 3. We extended a Choquet theorem on the construction of set-valued random variables to that on the construction of fuzzy set-valued random variables. That is, we showed existence of a fuzzy set-valued random variable associated with an alternating Choquet capacity of infinite order with respect to the graph topology which coincides the relative topology of product Fell-Matheron topology. 4. We gave a concrete metric compatible with the Fell-Matheron topology on the space of closed sets in a locally compact second countable Hausdorff space. We had to modify Hausdorff-Buseman which is given in the book "Random sets" by I. Molchanov introduced as a compatible metric. 5. We gave a strong law of large numbers for sums of independent fuzzy set-valued random variables, which are not necessarily indentically distributed.

In a joint work with Papasoglu, Fujiwara established a theory of "JSJ-decomposition" for finitely presented group along slender subgroups. The result is a far reach generalization of the theorem by Rips-Sela on the JSJ-decomposition along cyclic subgroups. A group is said "slender" if all of its subgroups are finitely generated. For example, any finitely generated abelian groups and nilpotent groups are slender, while a free group of rank at least two is not slender. The work is published in 2006.

We establish the following result : 1) Variational convergence of metric measure spaces: We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya. 2) Perturbation of symmetric Markov processes and its related stochastic calculus: We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang. 3) Kato class measures over symmetric Markov processes : We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi. 4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes: We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato. 5) Liouville theorems for harmonic maps to convex spaces over Markov chains: We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm. 6) Laplacian comparison theorem on Alexandrov spaces : We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya.

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.

Study of random variables taking values in general spaces might be an important theme both for theoretical and applied mathematics. One of the objects of this research is to study limit theorems for fuzzy sets-valued random variables. It is worth to note that the target space looses separability with some topologies. One of the results of this research is having noticed that laws of large numbers, central limit theorems and martingale convergence theorems hold with respect to the uniform topology with which the fuzzy set space is not separable. We used the method exploiting monotone property and that reducing to the theory of empirical distribution by proving the integrability of the entropy in the procedure to let the mesh smaller. Although the large deviation principles are more sensitive to the separability, we obtained Cramer type large deviation principles for as far as the topology induced by Levy's metric, and also for Skorohod topology and uniform topology with a little strong assumptions. We gave an explicit example which satisfies our assumptions. This seems to be a counter example to a result in a preprint appearing in an internet. We also obtained Sanov type large deviation principles under a natural assumption. Although we can compute rate functions explicitly only in simple cases, one of them is a relative entropy of two measures. Also, with the investigator H.Matsumoto, we obtained that a cM-X process is Markov only when c=0,1,and 2,where X is a one-dimensional Brownnian motion with constant drift and M is its maximum process. This is another part of Levy's theorem(for c=1) and Pitman's theorem (for c=2).

Let M_i→M and Y_i→Y, i=1,2,3,..., be two Gromov-Hausdorff convergent sequences of proper metric spaces, where ‘proper' means that any closed bounded set is compact. We give Radon measures on all M_i and M, and assume that the measure on M_i weakly converges to that on M. We are interested in the asymptotic behavior and convergence of maps u_i : M_i→Y_i. We introduce a concept of L^p-convergence of such u_i to a map u : M→Y,p【greater than or equal】1, and establish a theory of convergence of energy functionals E_i defined on the mapping space {u : M_i→Y_i} by generalizing Mosco's variational convergences. Mosco defined the asymptotically compactness of {E_i}, as a generalization of the Rellich compactness. The asymptotic compactness is useful to obtain the convergence of energy minimizers, i.e., harmonic maps. Under a uniform bound of the Poincare constant for E_i and some condition on the metric structure of M, we prove the asymptotic compactness of {E_i}. We say that E_i compactly converges to a functional E on {u : M→Y} if E_i Γ-converges to E and if {E_i} is asymptotically compact. We prove that the compact convergence E_i→E is equivalent to the Gromov-Hausdorff convergence of the E_i-sublevel sets to the E-sublevel sets. This gives a geometric interpretation of the compact convergence. Assume in addition that Y_i are all CAT(0)-spaces and E_i are convex and lower semi-continuous. Then, we prove that the compact convergence E_i→E is equivalent to the convergence of the corresponding resolvents, where the resolvents for E_i and E are defined by using the minimizers of the Moreau-Yosida approximation. As applications, we investigate the spectra of the Korevaar-Schoen approximating energy forms. We also obtain the compactness of the energy functionals over Riemannian manifolds under a bound of Ricci curvature.

The goal of this project is to study several problems on infinite discrete groups from the view point of geometry. "Geometric group theory" has its root in the pionear work by Gromov, and has been developed mainly in USA and Europe in the last 15 years or so. It is a dynamic field where one can apply classical combinatorial group theory, hyperbolic geometry, low dimensional topology, and the theory of mapping class groups. Unfortunately, there has not been much activities of this field in Japan yet. During the three years, we not only conducted our research, but also tried to put a foundation of the research activities of this field in Japan. One of our research themes has been on isometric actions of group on CAT(0) spaces. The notion of "CAT(0) spaces" was introduced by Gromov to geodesic metric spaces as a generalization of complete, simply-connected Riemannian manifolds, called "Hadamard manifolds". Given a discrete group G, it is important and useful to find a metric space X on which G acts on by isometries, properly. One classical example is the action of a lattice subgroup in a Lie group on its symmetric space. It would be interesting to find such X of minimal dimensions. In this direction, there has been a work by Brady-Crisp. We developed their work and found an answer to their question, found new examples, and formulated further questions in the paper "Parabolic isometries of CAT(0) spaces and CAT(0) dimensions", Fujiwara, Koji ; Shioya, Takashi ; Yamagata, Saeko. Algebr.Geom.Topol.4(2004), 861-892

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.

A crystal lattice is an abelian covering infinite graph of a finite graph. The integer lattices, the triangular lattice, the hexagonal lattice are examples of crystal lattices. We define magnetic transition operators to describe electron transfer on a crystal lattice under periodic magnetic field. The definition is justified by the central limit theorem : Namely, we show that the semigroup generated by the magnetic transition operators converges to the semigroup generated by a magnetic Laplacian of the Euclidean space with the Albanese metric. Magnetic fields on a crystal lattice are defined in terms of the second group cohomology. Next we construct a C^*-algebra associated with the magnetic field and show the magnetic transition operator belongs to the C^*-algebra. By using this, we show the spectra of the magnetic transition operators is a Lipschitz continuous function in magnetic field. Without magnetic field, electrons behave like random walks. We show large deviation principle holds for random walks on a crystal lattice. By letting lattice spacing smaller, a crystal lattice converges to a finite dimensional vector space with a Banach norm in the Gromov-Hausdorff topology. This Banach norm is characterized in terms of the rate function appearing in the large deviation.

(1)The study of variational convergence over metric measured space : This result is a joint work with Professor Takashi Shioya, who is an associate professor of Graduate School of Mathematical Institute, Tohoku University. We introduce a notion called asymptotic relation over a direct sum of metric spaces, which includes the notion of Gromov-Hausdorff convergence as an example. We discuss several notions of functionals over it, for example, Gamma convergence, Mosco convergence and compact convergence and so on. We give a sufficient condition for the compact convergence of functionals. We also prove a sufficient condition for the convergence of resolvents of energy functionals over CAT(0)-spaces. (2)The study on the stochastic representation of semigroups obtained from a non-symmetric perturbation : This study is a joint work with Professors P.J.Fitzsimmons, Z.Q.Chen and T.S.Zhang. Consider a symmetric regular Dirichlet form and the associated Hunt process admitting jumps of its sample paths. We consider a non-symmetric perturbation by use of locally square integrable martingale additive functionals and a continuous additive functional of finite variation. We prove that the corresponding semigroup has a stochastic representation in terms of time reverse operator on sample paths. (3)The study of Calabis type strong maximum principle : We give a stochastic proof of an extension of E.Calabi's strong maximum principle in the framework of strong Feller diffusion processes associated with local regular semi-Dirichlet forms. Our results can be applicable to the Gromov-Hausedorff limit space over a family of compact Riemannian manifolds with uniform lower bounds of Ricci curvature and uniform bounds of diameter.

T.Sakai, head investigator of this research program, has been working on the research theme : relationships between various metrical invariants of Riemannian manifolds, and their connection with the manifold structure. Under the support of the present Grant-in-Aid for Scientific Research, he especially studied the behavior of distance functions in Riemannian manifolds. 1. He begun to study the structure of Riemannian manifolds admitting a function f whose gradient is of constant norm under the project title "Curvature and structure of spaces" supported by the Grant-in-Aid for Scientific Research (C) (2), Nr. 09640109 (1997-1998). This is one of the remarkable properties of distance functions. He obtained characterizations of model warped product cases as equality case of inequalities in terms of the Laplacian of f, and investigated the perturbed version of the result, where the Ricci curvature played an important role. Under the support of the present Grant-in-Aid, he was engaged with the final step of this investigation. 2. Morse theory for a distance function on a Riemannian manifold : Although distance function f from a point p of a compact Riemannian manifold M admits points where f is not differentiable, it was known that the notion of critical points may be introduced as in usual Morse theory. However, the notion of the index of critical points of distance functions was not clear, and Sakai considered with J. Itoh the case where the cut locus C(p) of p carries a nice non-degeneracy structure. They showed in this case that the cut locus admits the Whitney stratification and developed Morse theory for distance function introducing the notion of the index of critical points. On the other hand, it later turned out that there are related works by V. Gerschkovich and H. Rubinstein, and we need more examination on the problem. Sakai gave a theme on "metrical invariants and the structure theorems on Alexsandrov spaces" to a student of doctor course and through examination some results related to the spheres were obtained. Sakai also worked for publication of survey articles "Curvature --Until the twentieth century, and the future? ", and "Family of Riemannian manifolds with Ricci curvature bounded below and its limits". 3. Research results of other investigators : Kiyohara determined the explicit structure of the cut locus of any point in ellipsoids. Katsuda studied the inverse problem of the Neumann boundary value problem, and Kasue investigated the spectral convergence of regular Dirichlet spaces including Riemannian manifolds, Riemannian polyhedra and sub-Riemannian manifolds. Shioya studied convergence and collapsing of Riemannian manifolds and spectrum of Laplacians. He also vigorously worked on geometry and analysis of Alexsandrov spaces. Tamura, studied Schroedinger operators and Dirac operators mainly from analytical viewpoint.

Many of our results are concerned with Riemannian geometry and the geometry of submanifolds. Spaces of constant curvature, such as spheres and Euclidean spaces, are the main models and reference spaces. We have greatly changed the models to wider classes of metrics. The Alexandrov-Toponogov comparison theorems for geodesic triangles on complete manifolds with base point at 0, whose radial curvature is bounded below by that of a model surface with rotationally symmetric metric have been established. Complete hypersurfaces with constant scalar curvature have also been investigated in details. The scaling limits of pointed complete open manifolds with asymptotically nonnegative radial curvature has been investigated.

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.

In this study, we construct foundation of analysis on Alexandrov spaces and develop it. We prove that the embedding of the (1, 2)-Sobolev space W^<1,2>(X) of a compact Alexandrov space X into the L^2 space is compact. As a corollary, we obtain the discreteness of the spectrum of the Laplacian on a compact Alexandrov space. Here, the Laplacian is defined as an infinitesimal generator of the energy form, by using functional analysis. For a DC function on an Alexandrov space we have another concept of Laplacian depending on DC charts, called the DC-Laplacian. We investigate the relation between the DC-Laplacian and the functional analytic Laplacian. Moreover, we prove the continuity of the solutions of the eigen-equation and the heat equation on Alexaudrov spaces, and also the existence of the heat kernel. Let A(n) be the set of compact Atexandrov spaces of dimension n and curvature 【greater than or equal】 -1. We prove that on A(n) the spectral topology due to Kasue-Kumura coincides with the Gromov-Hausdorff topology. We introduce a concept of 'spaces of Ricci curvature bounded below' and energy of maps from such a space to a complete metric space. We prove the Poincare inequality and the existence of energy measure for it.

In this research from 1998 to 2000, we have decided the topology of collpsed Riemannian closed 4-manifolds with uniform lower bound on sectional curvature and upper diameter bound in terms of the singular fiber structure over the limit Alexandrov space. In the course of the proof of this result, we have also succeeded the classification of complete nonnegatively curved Alexandrov spaces of ditmensions three and four. The detail of the results are the following : ・We have constructed a local S^1-action on a collapsed 4-manifold when the limit space is of dimension three ; ・We have proved that a collapsed 4-manifold is either a sphere-bundle or a Seifert torus-bundle over the limit space ; ・In the case when the limit has dimension two and non-empty boundary, a decomposition of the collapsed 4-manifold along the boundary of the limit space has been obtained ; ・In the case when the limit is an interval, we have proved that the collapsed 4-manifold is a gluing of at least two and at most four disk-buundles ; ・We have proved a soul theorem for 4-dimensional concompact complete Alexandrov spaces with nonnegative curvature ; ・We have succeeded in proving a splitting theorem for complete Alexandrov spaces with nonnegative curvature and with disconnected boudary ; ・We have obtained a metric classification of compact Alexanadrov spaces with nonnegative curvature and with maximal vertices.

・6月のOverwolfachで、Colding氏(New-York)が,リッチ曲率【greater than or equal】-1なるn次元Riemann多様体の列M_iが距離空間XにGromov-Hausdorff距離に関して収束するとき、M_iのラプラシアンがXのラプラシアンに収束すると発表した。これは、dimX=nの場合と思われる。この研究集会でのColding氏との議論は本研究に大変有益であった。 ・断面曲率【greater than or equal】-1の場合には、今年度分担者塩谷との共同研究により、3次元における崩壊が明らかになった。従って、今後これをリッチ曲率【greater than or equal】-1の場合に拡張してラプラシアンのコンパクト化 ・Alexandrov空間上のラプラシアンに関して国内で進展があった。塩谷が,佐賀大の桑江・町頭両氏との共同研究により、Alexandrov空間上のラプラシアンと熱核を構成した。これにより、特異リッチ曲率をもつ空間上のソボレフ型埋め込み定理を得る事がより現実的な課題となった。 ・分担者塩浜との特異空間の幾何学に関する有益な議論、国内の研究集会への出席や許洪偉氏(中国)の九大への招聘などによる大域解析学に関する有益な議論,等を通して本研究の確かな方向づけや深い知見が得られた。 ・幾何学・大域幾何学を中心とした書物の購入により,その基礎概念から最新の理論まで,てっとり早く仕入れることが出来き、また、大画面のコンピュータ・デスプレイの購入により、論文書きがよりスピーデイに行え本研究遂行に役立った。

本研究では曲率の積分が一様に有界な曲面の極限空間の位相構造についての結果を得た。その様な空間は(位相的に)多様体にはならず、局所2連結でさえないが、それでも位相的構造を完全に解明した。正確には以下のように述べられる。球面を数珠繋ぎに(有限または無限個)繋げた空間をa string of pearlsと呼ぶ。位相空間Xが、その任意の点のある近傍がstring of pearlsを有限個接着したものに同相となるとき、Xをpearl spaceと呼ぶことにする。このとき、与えられた位相空間Xに対して、次の2つは同値になることを証明した。 1.Xにある距離構造が存在して、Xは全曲率が一様にな閉曲面の極限となる。 2.Xはpearl spaceである。 これを証明するためにToponogovタイプの比較定理を曲率の積分が小さい場合に成り立つことを証明し、それを使って全曲率が一様に有界な曲面の極限空間上の幾何学を確立した。上の結果はこれら全ての研究の最終的帰結として得られたものである。

山口孝男はアレクサンドロフ空間を一般化した局所的リプシッツ可縮内部距離空間X上のリプシッツ鎖の体積に関する理論を構成し、その応用として曲率を上から制限したアレクサンドロフ空間上の強意m-単体のハウスドルフ測定の上からの評価を得た。この結果はGromovによるRic≧n-1なる条件下での単体体積に関する評価を発展させたものである。塩谷隆は全絶対曲率の上限をもつ2次元コンパクトリーマン多様体族は、ハウスドルフ距離に関してprecompactである事を示し、その閉包に属する内部距離空間Xのハウスドルフ次元に関する上からの精密な評価を得た。かようなXはアレクサンドロフ空間の性質を持たず、π_2(X)が無限生成となる例も含み、極めて興味深い独創的な結果である。塩浜勝博は許洪偉と共同で第2基本形式及び曲率ノルムと位相との関係を研究した。Nash embeddingによって次元の高いユークリッド空間に等長的にうめ込まれた多様体族を第2基本形式から定まる量によって定め、球面定理を証明した。曲率ノルムに関しても同様の結果を得た。これらの条件をみたす多様体族はアレクサンドロフ空間の範疇に属するが、曲率の上下からの制限が完全に撤廃された。かくして全く新しいリーマン多様体族に関する球面定理がうち立てられた。塩浜は更に極小カレントに関数するFederen-Flemingの定理とLawgon-Simonsの結果を用いて、ホモロジー消滅定理を第2基本形式から定まるスカラー評価から示し、極小部分多様体論がリーマン幾何学にも適用出来る初めての例となる成果を挙げた。これら全ての結果はアレクサンドロフ空間の幾何学の発展の方向を示す有力な手がかりを与える貴重なものと言えよう。

In the case where the singularities of Alexandrov spaces with curvature bounded below are not so big, under convergence of spaces. we were able to construct Lipschitz homeomorphisms between spaces. In particalar, the continuity of volumes of Alexandrov spaces follows from this result. Moreover, we proved that the Hausdorff measure of the singular set of an Alexandrov space is zero, and that one can define a natural Riemannian structure on the regular set. We also proved that the isometry group of an Alexandrov space with curvature bounded below is a lie group, which has some applications to Riemannian geometry. On the other hand, we extended the notion of the Gromov invariant to Alexandrov spaces, and clarified the relation between the curvature, volume and the Gromov invariant. First, making use of the Alexander-Spanier cohomology theory, we proved the existence of the fundamental class [X] of X, and defined the Gromov invariant of X.Next, we proved that the mass of the fundamental class [X] coincides with the volume of X.In the proof of this face, we used geometric measure theory to approximate a chain representing [X] in the mass topology by a Lipschitz chain with nice properties, and developed a cancellation technique which might be considered as a replacement of Stokes' theorem. And we proved that the Gromov invariant of a negatively curved Alexandrov space can be estimated below interms of the upper bound of curvature and the volume. In the case of Alexandrov surfaces, we obtained a sharp estimate for the Gromov invariant with the type of singularities. For Alexandrov spaces with curvature bounded below, we bave an estimate for the Gromov invariant from above in terms of the volume and the lower bound of curvature. Thus it turned out that the appearanceo of singularities of such a space does not affect the Gromov invariant so much. This shows the big difference between the two cases, spaces curved above and spaces curved below.

X,Yを群Gに値をとる独立確率変数列で,Xは同分布列,またX・Yを項別積列とする。今XとX・YがG値列全体の空間に導く確率測度をそれぞれPとQとするときPとQの絶対連続性(P〜Qと書く)をYの分布で特徴づけることが問題であった。この問題の最も簡単で基本的な場合がGが実数加群の場合であり,これについては(代表者佐藤の貢献を含め)多くの結果が知られている。今年度はまずGがこの実数加群の場合について知られていることを整理することから始めた。これらをまとめて「無限次元確率解析」日米シンポシウム(米国・バトンル-ジュ市)で発表し,好評を博した。 次にGが正の実数のなす乗法群の場合に,Xの1次元分布がルベ-グ測度と互いに絶対連続であること,その密度関数がある種の積分正測性をみたすこと,を仮定することによって,Yの平均が2乗総和可能列であることがP〜Qとなるための十分条件であることを示した。しかしこの場合は対数写像を考えるとGが実数加群の場合と本質的には同じであると考えられる。 またXが実確率変数列でYが正値確率変数列の場合にもP〜Qとなるための十分条件が得られた。しかしこのような簡単な場合でも必要条件はかなり難しいことが分かった。 他方,ユークリッド空間上の局所可積分関数の最大関数について,(p,p)強有界性が次元に無関係に得られることがStein & Stroembergによって証明されている。これは無限次元ガウス測度の密度定理を考える上で大いに参考になりそうな結果がある。ところがその証明は大変難解である。そこでまず今年度はこの結果に完全証明をつけることから始めた。その成果は博士前期課程大塚正治君の修士論文としてまとめた。 さらに分担者達もそれぞれの立場から成果を得た。

横田佳之:トンネル数1の結び目の2つの連結和のトンネル数が3となることがあることを発見し,さらに,トンネル数の評価法を研究した。(佐久間誠,森元勤治との共同研究) 坂内悦子:スピンモデル,アソシェーションスキームの相対性及びモジュラー不変性の相互関係を研究し,ある種のアソシェーションスキームからスピンモデルを構成するアルゴリズムを見い出し,それを有限群から得られるアソシェーションスキームに適用して,そのモジュラー不変性を完全に決定し,さらに,非対称スピンモデルの構成に成功した。(坂内英一,F.Jaegerとの共同研究) 山田美枝子:ジョーンズ型対称スピンモデルの4つのうち2つが,ある4-weight スピンモデルから構成できること,また,4-weight スピンモデルがある条件をみたせば、その2倍のループ変数をもつ,しかも,その条件をみたす4-weight スピンモデルが構成できることを示し,実際,アダマ-ル行列からこの構成法により,その無限系列を構成した。 塩谷隆:下に曲率が有界な距離空間(アレキサンドロフ空間Xの特異集合S_Xの余ハウスドルフ次元が1以上であることを示し,X-S_Xにリーマン構造が入ることを示した。 川崎英文:動節点をもつスプライン関数によるチェビシェヲ近似問題を最適化問題として捉えると,1次の最適性条件をみたせば,2次の最適性条件をみたすという特殊な構造があることを発見した。

完備でR^2に同相なリーマン多様体をリーマン平面と呼ぶ。リーマン平面の全曲率はCohn-Vossenの定理により2pi以下であることが知られている。以前の研究において、リーマン平面内の極大測地線の振舞いについて全曲率が2piより小さい場合に調べた。ここで極大測地線とはR上で定義された一点でない測地線である。今年度の研究では全曲率が丁度2piのとき、および境界を持つ2次元リーマン多様体の測地線の振舞いについて結果を得た。この研究において理想境界の概念は非常に重要である。具体的には次を証明した。全曲率が2piのリーマン平面をMとするとき、任意の数nに対して、Mのあるコンパクト集合K_nが存在し、K_nの外側にある任意のMのproperな極大測地線に回転数がn以上の部分弧が存在する。ここで測地線がproperとはRからMへの写像としてproperということであり、回転数とはwhitneyが定義した位相的な回転数である。さらに、Mの全ての閉測地線がある一つのコンパクト集合に含まれるような全曲率をもつリーマン平面の極大測地線の振舞いについて調べた。このようなMはexpanding,contractingと呼ばれる2つのクラスに分類される。Mがcontractingのとき、Mのある十分大きなコンパクト集合の外側には極大測地線は存在しない。一方Mがexpandingのとき、そのような極大測地線は必ず存在し、その位相的な形は決定される。