2018年度年会(於:東京大学)

SUMMARY: We discuss countable groups, mainly focusing on problems related to existence of (bounded) harmonic functions. In particular we study questions such as which group does not admit any non-constant bounded harmonic function (Liouville property), and when it admits such a function, how all such functions are obtained (Poisson boundary). This problem often requires deep understanding on geometry of underlying groups as well as quantitative behavior of random walks. I will try to present this subject with several explicit key examples, emphasizing on importance of combining different ideas and techniques.

SUMMARY: Since the Seiberg–Witten equations were introduced by Witten in 1994, the equations have produced many significant applications to 3 and 4 dimensional geometry. In this talk, we will discuss some of them and recent progress.

SUMMARY: In this talk, I will overview the recent progress on my research on the complex analysis on Teichmüller space. The aim of this research is to give a unified treatment between the topological aspect and the complex analytical aspect in Teichmüller theory. I will discuss the infinitesimal deformation of singular Euclidean structures on a surface in aiming for developing the Teichmüller geometry (Extremal length geometry) on Teichmüller space. I also give a formula of the Levi form of the Teichmüller distance and the pluricomplex Green function on the Teichmüller space. If time permits, I will give an idea for unification and a conjecture on the pluricomplex Poisson kernel on the Bers slice.

SUMMARY: The existence of the curve to give the inverse linear parallel displacement is not known for a linear parallel displacement of Finsler space generally. A purpose of the study is to establish a mathematical method to find such a curve, but I don’t have it, yet. Today, I report a necessary and sufficient condition for two curves that they are to be such curves to give other inverse linear parallel displacement each other.

SUMMARY: Symplectic-Haantjes manifolds are constructed for several Hamiltonian systems following Tempesta–Tondo, which yields the complete integrability of systems.

SUMMARY: A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on \(S^4\) associated with a holomorphic Poisson structure on \(\mathbb {CP}^3\). The space of the Poisson structures on \(S^4\) is a real algebraic variety in the space of holomorphic Poisson structures on \(\mathbb {CP}^3\). We generalize it to \(\mathbb {HP}^n\) by using the twistor method. Furthermore, we provide examples of Poisson structures on \(S^4\) associated with codimension one holomorphic foliations of degree 2 on \(\mathbb {CP}^3\).

SUMMARY: A complex contact structure \(\gamma \) is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle \({\rm Ker}\, \gamma \) of the tangent bundle and a line bundle \(L\). In this paper, we prove that the sheaf of holomorphic \(k\)-vectors on a complex contact manifold splits into the sum of \(\mathcal {O}(\bigwedge ^{k} {\rm Ker}\, \gamma )\) and \(\mathcal {O}(L\otimes \bigwedge ^{k-1} {\rm Ker}\, \gamma )\) as sheaves of \(\mathbb {C}\)-module. The theorem induces the short exact sequence of cohomology of holomorphic \(k\)-vectors, and we obtain vanishing theorems for the cohomology of \(\mathcal {O}(\bigwedge ^{k}{\rm Ker}\, \gamma )\).

SUMMARY: Belgun proved that Inoue surface has no Vaisman structures. In this talk, we generalize this result and construct solvmanifolds without LCK structures. Note that these solvmanifolds have LCS structures.

SUMMARY: In the last 20 years, the Hermite–Liouville structures on compact complex manifolds have been studied. In these studies, almost all of the non-Kählerian structures were obtained by deforming from the Kähler–Liouville structure. In this presentation, the speaker will illustrate the construction of the examples of the Hermite–Liouville structure on the Hopf surface, which leads to the complete integrability of its geodesic flow.

SUMMARY: Ricci Calabi functional is a functional on the space of Kähler metrics of a Fano manifold. Its critical points are called generalized Kähler Einstein metrics. In this talk, we show that the Hessian of the Ricci Calabi functional is non-negative at a generalized Kähler Einstein metric.

SUMMARY: We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano manifolds.

SUMMARY: In order to study the optimal degeneration of a Fano manifold, we introduce the Ding flow as the gradient flow of the Ding energy functional on the space of Kahler metrics.

SUMMARY: I will talk about one of coarse geometric versions of the so-called Cartan–Hadamard theorem, that is, a coarse Cartan–Hadamard theorem on coarse convex spaces. Also I will deal with coarse homotopy, open cones and the coarse Baum–Connes conjecture. This talk is based on a joint-work with Tomohiro Fukaya (Tokyo Metropolitan University); ‘Tomohiro Fukaya, Shin-ichi Oguni, A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture, preprint, 2017, arXiv:1705.05588’.

SUMMARY: In the joint work with Shin-ichi Oguni, we introduced a new class of metric spaces which we call “coarsely convex spaces”. This is a new formulation of “nonpositively curved spaces” from the view point of coarse geometry. This class includes Gromov hyperbolic spaces, CAT(0)-spaces, and systolic complexes. This class is closed under quasi-isometry, and direct product. The idea of the definition is “convexity of metric” and its coarsification. We also construct an ideal boundary for coarsely convex spaces. The construction is based on that of Gromov hyperbolic spaces. Due to time constraints, in this talk, we will not explain on an application to the coarse Baum–Connes conjecture, which is the original motivation of this work.

SUMMARY: In this talk, we present the result that the \(\mathrm {Cycl}_4 (0)\) condition implies the \(\mathrm {Cycl}_k (0)\) condition for any integer \(k\geq 4\). We also present the result that a five-point metric space embeds isometrically into a \(\mathrm {CAT}(0)\) space if and only if it satisfies the \(\boxtimes \)-inequalities.

SUMMARY: Gigli, Mondino, and Savaré introduced the pmG-convergence on the space of pointed metric measure spaces and studied the stability of the curvature-dimension condition and the Mosco convergence of Cheeger energies under the pmG-convergence. We introduce a new condition generalizing the pmG-convergence and then prove similar results under this condition. Our study is also related to the study by García, Kell, Mondino, and Sosa for quotient spaces by actions of compact groups.

SUMMARY: We prove that if a geodesic metric measure space satisfies a comparison condition for the isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger–Gromall’s splitting theorem and also of Cheng’s maximal diameter theorem. As an application, we obtain an isometric splitting theorem for a complete weighted Riemannian manifold with positive Bakry–Émery Ricci curvature.

SUMMARY: M. Gromov introduced the Lipschitz order relation on the set of metric measure spaces and developed a rich theory. For a metric measure space \(X\), we consider the set of the distributions of 1-Lipschitz functions on \(X\) and we call it the 1-measurement of \(X\). We also define Lipschitz order on the 1-measurement naturally. The existence of the maximum of 1-measurement is deeply related to the isoperimetric inequality of \(X\). In fact if \(X\) is an \(n\)-dimensional sphere, we obtain the maximum of 1-measurement by the isoperimetric inequality. However, if \(X\) is a \(n\)-dimensional Hamming cube, the maximum of 1-measurement does not exist because of discreteness. We solve this problem by generalizing the definition of Lipschitz order with an error. On the spheres case, we have Normal law à la Lévy by considering the weak limit. We have the Hamming cubes version of it as an application of the main theorem.

SUMMARY: We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.

SUMMARY: We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below, defined in terms of comparison angles. We first find relations between this invariant and volume. We discuss the case of maximal obtuse constant equal to \(\pi /2\), where we prove some rigidity for spaces. Although we consider Alexandrov spaces with curvature bounded below, the results are new even in the Riemannian case.

SUMMARY: In the history, there are several elementary theorems by figure. We try to find such theorems, and found new 6 theorems. We show entire structures in figures of our text. We explain briefly them by naming as (1) star-star theorem (2) quadrangle Stainer theorem (3) 6 perpendicular-lines theorem (these 3 are concurrent theorems), (4) 10 lines theorem (5) 2 circles system theorem (6) Hexagon Theorem (these 3 are Collinear theorems). These are not proved. But, these theorems include important theoretical structures, and, it is interesting to follow the drawing orders and to consider on the compositions. Anyway, we show all figures of Theorems in our text. Please enjoy many strangeness of theorems. We will be able to speak some relations of theorems in later.

SUMMARY: We show that the intrinsic volumes of compact bodies in the Euclidean spaces of dimension two and three can be obtained from the residues of the (relative) Brylinski beta functions.

SUMMARY: A. Pelayo–A. R. Pires–T. S. Ratiu–S. Sabatini defined a metric on the set of Delzant polytopes. They studied structures of the metric space and the moduli space with respect to the action of the integral affine transformations. The definition of the metric is natural, though, it does not induce a metric on the moduli space. In this talk we would like to try to define a metric on the moduli space. We can show that it actually defines a metric on the moduli space for 2-dimensional case.

SUMMARY: We consider orthocenters of simplices of hyperbolic spaces. Unlike the cases of Euclidean spaces or spheres, the similar condition does not always imply the existence of orthocenters. In this talk, we give characterizations of the existence of orthocenters.

SUMMARY: The moduli space of left-invariant metrics on a Lie group is defined as the orbit space of the action of the group of automorphisms and scalings on the space of left-invariant metrics, and has been studied actively. In this talk, we focus on some kinds of singular points which arise in the moduli space. We show that if an equivalent class of a left-invariant metric is the “most singular” point in the moduli space, then the left-invariant metric has nice properties.

SUMMARY: We show that a link in a closed orietable 3-manifold can be realized as the set of complex tangents of a smooth embedding of the 3-manifold into the complex 3-space if and only if it represents the trivial integral homology class in the 3-manifold.

SUMMARY: We prove that in any 2-torus \(T^2\) for any point \(p \in T^2\) and for any \(\varepsilon > 0\) there exists a number \(R > 0\) such that the geodesic circles with center \(p\) and radii \(t\) are \(\varepsilon \)-dense in \(T^2\) for all \(t > R\).

SUMMARY: We prove that for an arbitrarily given compact Riemannian manifold \(M\) admitting a point \(p \in M\) with a single cut point, every compact Riemannian manifold \(N\) admitting a point \(q \in N\) with a single cut point is diffeomorphic to \(M\) if the radial curvature of \(N\) at \(q\) are sufficiently close in the sense of \(L^1\)-norm to that of \(M\) at \(p\).

SUMMARY: We will consider the upper bound for the first eigenvalue of the Laplacian on a closed surface. For the genus two case, we obtain a singular metric which maximize the first eigenvalue. This result was conjectured by Jakobson–Levitin–Nadirashvili–Nigam–Polterovich. It is joint work with Shin Nayatani, Nagoya University.

SUMMARY: Ghosh and Sharma have studied \((\kappa ,\mu )\)-spaces, which are contact metric spaces with certain nullity conditions. Especially, they gave a necessary condition for \((\kappa ,\mu )\)-spaces to be nongradient Ricci soliton manifolds. In this talk, we prove that in the connected, simply-connected and complete case such \((\kappa ,\mu )\)-spaces can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians. Consequently, we also prove that such spaces are actually Ricci soliton.

SUMMARY: Let us consider an \(n\)-dimensional complex torus whose period matrix is \((I_n,T)\). Here, \(I_n\) is the identity matrix of order \(n\) and \(T\) is a complex matrix of order \(n\) whose imaginary part is positive definite. In particular, when we consider the case of \(n=1\), i.e., a one-dimensional complex torus, the corresponding complexified symplectic form of the mirror partner of the one-dimensional complex torus is defined by \(-\frac {1}{T}\) or \(T\). However, if we assume \(n \geq 2\) and that \(T\) is a singular matrix, we can not define the mirror partner of the complex torus as the natural generalization of the case of \(n=1\) to the higher dimensional case. In this talk, we propose a way to avoid this problem, and discuss the homological mirror symmetry.

SUMMARY: We define a class of harmonic Hadamard manifolds of hypergeometric type. This class of harmonic manifolds includes all Damek–Ricci spaces and also all rank one symmetric spaces of non-compact type as particular cases. Using a hypergeometric description of spherical functions on each harmonic Hadamard manifold \(X\) belonging to this class, we discuss harmonic analysis of radial functions on \(X\). In this talk we would like to present the inversion formula, Plancherel theorem and Paley–Wiener type theorem for the spherical Fourier transform on a Hadamard harmonic manifold which is of hypergeometric type.

SUMMARY: In this talk I consider a homogeneous holomorphic line bundle over a certain elliptic (adjoint) orbit, and set a representation of real semisimple Lie group on a complex vector subspace of the complex vector space of holomorphic cross-sections of the bundle. Then, I state that the representation is irreducible unitary.

SUMMARY: Let \(X\) be a smooth compact manifold with corners which has two embedded boundary hypersurfaces \(\partial _0 X , \partial _1 X\), and suppose a fiber bundle \(\phi : \partial _0 X \to Y \) is given. We define a pseudodifferential calculus \(\Psi ^*_{\Phi ,b}(X)\) generalizing the \(\Phi \)-calculus of Mazzeo–Melrose and the \(b\)-calculus of Melrose. We investigate the Fredholm condition and the index of an operator \(P \in \Psi ^*_{\Phi ,b}(X)\). And as its application, we prove the index theorem of “non-closed” \(\mathbb {Z}/k\)-manifolds.

SUMMARY: A wave equation of motion of an elastic wire on a Riemannian manifold has a solution for any initial data.

SUMMARY: We generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler–Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form \(\omega \in 2\pi c_1(M)\). Namely, we consider a variational problem for Lagrangian submanifolds in a Kähler manifold \(M\) w.r.t. a weighted volume functional. Moreover, we introduce the generalized Lagrangian mean curvature flow in a Fano manifold, and we show that if the initial Lagrangian is a small Hamiltonian deformation of a minimal and Hamiltonian stable Lagrangian w.r.t. the weighted volume functional, then the generalized MCF converges exponentially fast to a minimal Lagrangian submanifold.

SUMMARY: An anisotropic surface energy is a generalizasion of the area of surfaces. It is the integral of an energy density function which depends on the surface normal over the considered surface, and it serves as a mathematical model of energy of crystals. The absolute minimizer of an anisotropic surface energy functional among all closed surfaces enclosing the same volume is unique and it is called the Wulff shape. In this talk, we show that, if the energy density function is not “convex”, there exist closed equilibrium surfaces of the anisotropic surface energy for volume-preserving variations which are not the Wulff shape. By applying this result, it is shown that the uniqueness for closed self-similar solutions with genus zero for anisotropic mean curvature flow does not hold in general. These concepts and results are naturally generalized to higher dimensions.

SUMMARY: An anisotropic surface energy is a generalizasion of the area of surfaces. It is the integral of an energy density function which depends on the surface normal over the considered surface, and it serves as a mathematical model of energy of crystals. The absolute minimizer of an anisotropic surface energy functional among all closed surfaces enclosing the same volume is unique and it is called the Wulff shape. In this talk, we show that, if the energy density function is of \(C^3\) and “convex”, then any stable closed equilibrium surface of the anisotropic surface energy for volume-preserving variations is (up to homothety and translation) the Wulff shape, here an equilibrium surface is said to be stable if the second variation of the energy for all admissible variations is nonnegative. The result holds also for hypersurfaces in any Euclidean space.

SUMMARY: We discuss transforms for minimal surfaces in 5-dimensional Riemannian space forms, and Lorentzian minimal surfaces in the 5-dimensional semi-Euclidean space of index 2.

SUMMARY: We give transforms and a representation formula for non-conformal harmonic surfaces in the Euclidean 3-space.

SUMMARY: In this talk, we give a proof for calibrated equalities from the viewpoint of group actions. The essential part of our proof is to describe the orbit spaces for certain group actions on oriented Grassmann manifolds.

SUMMARY: In this talk, we give a proof for Mealy’s calibrated inequalities from the view point of group actions. As a typical example, we prove Wirtinger’s inequality in terms of the duality which was introduced by the speakers.

SUMMARY: For a regular surface in Lorentz–Minkowski 3-space, a point is called a lightlike point if the first fundamental form is degenerate at the point. In this talk, we prove that any analytic surface admits non-trivial isometric deformations around a non-flat and non-degenerate lightlike point.

SUMMARY: In this talk, we will classify austere submanifolds in compact simple Lie groups. In particular, we proved that Cartan embeddings which defined by inner automorphisms of finite order \(k >2\) of compact simple Lie gruops are not austere.

SUMMARY: An adjoint orbit \(M\) of a compact connected semisimple Lie group \(G\) is called a complex flag manifold. The intersection of two real forms \(L_0\) and \(L_1\) in a complex flag manifold \(M\) is an antipodal set of \(M\). Applying the antipodal structure of the intersection \(L_0 \cap L_1\), we calculate the Lagrangian Floer homology \(HF(L_0, L_1 : {\mathbb Z}_2)\), when \(M\) has a \(G\)-invariant Kähler–Einstein metric and when two involutions of \(G\) defining \(L_0\) and \(L_1\) commute with each other.

SUMMARY: We construct a pseudo-Anosov automorphism whose dilatation is a 2-Salem number by means of the spectrum radius of the bicolored Coxeter element of a bipartite Coxeter system.