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

Periodic minimal surfaces in the Euclidean three-space can be considered as a mathematical model for surfactant in the soft matter. In 1990s, physicists considered many families of triply periodic minimal surfaces. On the other hand, a stability of a minimal surface has been studied via area minimizing situation. In particular, Barbosa-doCarmo's technique related to volume preserving stability might be useful tool for this situation. By this study, we find out volume preserving stability for some families which physicists considered. Moreover, we gave mathematical description of lamellar phases in the soft matter as limits of triply periodic minimal surfaces.

Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）

By the arithmetic Schottky-Mumford uniformization theory, we proved the arithmeticity of Chern-Simons invariants. Using this result together with the Arakelov theory in arithmetic geometry and the theory of Zograf, Mcintyre-Takhatajan on the classical Liouville field theory, we gave an infinite product presentation of the Deligne-Riemann-Roch isomorphism which expresses the Chern-Simons line bundle. As its application, we express the special values of the Ruelle zeta functions of Schottky groups as the products of period integrals and discriminants. This result gives an analog of the Deligne conjecture on
such geometric zeta values.

Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）

We studied the projective dimension of symbolic powers of squarefree monomial ideals in a polynomial ring. We proved that the projective dimension of the symbolic power of the edge ideal of a very well-covered graph increases with respect to the exponent. Since a well-covered bipartite graph is very well-covered and since the symbolic powers and ordinary powers coincide for the edge ideal of a bipartite graph, it implies that the projective dimension of the ordinary power of the edge ideal of a very well-covered graph increases. Moreover, we showed that the projective dimension of the symbolic power of the edge ideal of a graph with a vertex of degree one increases with respect to the exponent.

Grant amount：\10790000 （ Direct Cost: \8300000 、 Indirect Cost：\2490000 ）

Under given boundary conditions, minimal surfaces are critical points of area, and surfaces with constant mean curvature (CMC surfaces) are critical points of area among surfaces enclosing the same volume. Such a surface is said to be stable if it attains a local minimum of area for all admissible variations. In this research, we studied criteria for stability, existence and uniqueness of (stable) critical points, bifurcation of critical points, for fixed, free, and periodic boundary conditions.

Grant amount：\2990000 （ Direct Cost: \2300000 、 Indirect Cost：\690000 ）

It was investigated about the educational effect of the attached school in national university. That was chosen as the way to interview students and interpret their obtained talk (history of learning), not achievement tests.
It's the case that students' learning is embedded in the situation of the school and the home that it became clear. Therefore learning is restricted by the situation-pressure by which it's for the school and the home. On the other hand they get a resource from situations and are expanding learning aggressively. Therefore the education which emphasized the situation of the school and the home is desired.

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

We would study periodic minimal surfaces in the Euclidean space in terms of the differential geometry. In particular, we treated Morse indices of triply periodic minimal surfaces. In this period, we could succeed in computing many Morse indices for families of minimal surfaces which have been studied in physics and chemistry. Also, we could succeed in mathematical description of Lamellar structure by limits of triply perioddic minimal surfaces. Moreover, we obtained the existence and uniqueness results for a comlete minimal surface of finite total curvature in the Euclidean space.

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,

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.

Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）

We usually call the set of all minimal surfaces" Moduli space of minimal surfaces". To consider the Moduli space of minimal surfaces, it is important to study Period map on the Moduli space of minimal surfaces. We have calculated periods of many examples of minimal surfaces. From the experiences, we conjecture that there may be symmetry by solvable group in Galois Theory. Thus, we will establish the Moduli theory of minimal surfaces in terms of the Galois Theory.

Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

In this project, by making use of fusion on the research methods in the geometry of submanifolds and the research methods for eigenvalues of the differential operators on bounded domains in Riemannian manifolds, we can construct trial functions with good properties such that we obtain a sharp universal inequalityfor eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domainsin Riemannian manifolds and a universal inequality for eigenvalues of the clamped plate problem. Furthermore, by combining the universal inequality for eigenvalues with therecursion formula of Cheng and Yang, we give the upper bounds for the k-th eigenvalueof the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. It is optimal in the sense of the order of k. By using a new and original method replacing the method of the Fourier transform, we obtain a Li-Yau type lower bound for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. About the study on lower bounds for eigenvalues of the clamped plate problem on bounded domains in a Euclidean space, we improve the inequality of Levine and Protter by using the Fourier transform and the symmetry rearrangement of a domain. On the other hand, we study structures of curvatures and topological structures of submanifolds according to several different view points. An optimal upper bound for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in the unit sphere is given. Many embedded compact hypersurfaces with constantk-th mean curvature in the unit sphere are constructed.

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.

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.

Grant amount：\1100000 （ Direct Cost: \1100000 ）

本年度の研究課題はSpecial Lagrangian多様体を幾何学的測度論の観点から研究する事であった.そのために体積最小に関する古典的な部分多様体である極小曲面の研究方法をSLの研究へと適応する事を試みた.その前段階として平坦トーラス内の極小曲面に関する研究を行った.
まず4次元平坦トーラス内の極小曲面において,筆者による先行研究の具体例よりも種数の高い具体例を構成した.今回構成した具体例は長野-Smythによる結果の逆が一般には成り立たない事を示すのみならず,極小曲面全体のモジュライの研究に関係するものである.また,その後イギリスのWarwick大学のスタッフであるM.Micallef氏のもとに滞在し,この具体例の指数に関する研究を行った結果,それは6以上である事が判った.次に6次元平坦トーラス内の安定極小曲面でトーラスのどのような複素構造に対しても複素正則にならないような安定極小曲面の存在に関する研究を行った.この問題はC.Arezzp, Micallef, R.Schoenをはじめとして多くの研究者の興味をひくものであり,未だ解決されていない問題である.その第一歩としてまず具体例の構成を考えた.構成の手段として,5次元平坦トーラス内の極小曲面からそれを1助変数にて変形して6次元平坦トーラス内の複素正則な極小曲面を構成した.現在,その複素正則な極小曲面に近い極小曲面における安定性を考察中である.さらに,神戸大の藤森祥一氏との共同研究にて,筆者による3次元平坦トーラス内の極小曲面の具体例のグラフィックスおよびその構造が明らかになった.これは6本の閉折れ線によるPlateau問題の解となる極小曲面を鏡像の原理で増やしたものである.残念ながら本研究が目指す結び目に関するPlateau問題による構成ではないものの十分にその価値を評価しうるものである事が判った.