2019年度秋季総合分科会(於:金沢大学)

SUMMARY: It is known (by B. Palmer) that for each oriented hypersurface \(M^n\) in sphere \(S^{n+1}\), the image of the Gauss map \(\gamma \) of \(M\) into complex \(Q^n\) is a Lagrangian submanifold. Moreover if \(M^n\) is isoparametric, then \(\gamma (M)\) is a minimal Lagrangian submanifold in \(Q^n\). We define the Gauss map \(G\) for real hypersurfaces \(M^{2n-1}\) in complex projective space \(CP^n\), into complex \(2\)-plane Grassmannian \(G_2(C^{n+1})\). If \(M\) is a Hopf hypersurface in \(CP^n\), then the \(\gamma (M)\) is a half dimensional ‘totally complex submanifold’ in \(G_2(C^{n+1})\) with respect to the quaternionic Kähler structure. Hence each Hopf hypersurface in \(CP^n\) is a total space of circle bundle over a Kähler manifold \(\gamma (M)\). Also we have ‘converse construction’ by using the twistor space of \(G_2(C^{n+1})\). We have similar results for real hypersurfaces in complex hyperbolic space \(CH^n\).

SUMMARY: In the late 1950’s Kolmogorov discovered that Shannon’s entropy can be used in ergodic theory. This is a revolutionary idea, and ever since there have been rich interactions between information theory and the study of dynamical systems. Recently we have added some new items in these interactions. A new development comes from mean dimension theory. Mean dimension is a topological invariant of dynamical systems which estimates the number of parameters per iterate for describing the orbits of dynamical systems. We have found that this dynamical invariant has the following two connections with information theory:

(1) Mean dimension turns out to be a crucial parameter when we try to encode dynamical systems into band-limited signals, say signals of telephone line. This is reminiscent of Shannon’s fundamental work on communications over band-limited channels. This discovery was used to solve a problem posed by Lindenstrauss in 1999.

(2) Mean dimension theory is (in some sense) a topological version of rate distortion theory. Rate distortion theory is a branch of information theory describing a lossy data compression method achieving some distortion constraint. We study the minimax problem about the “rate distortion dimension” and shows that the minimax value is given by mean dimension at least for minimal dynamical systems. This is a mean dimensional analogue of variational principle known for dynamical entropy.

SUMMARY: I will talk about symplectic capacities, in particular those related to periodic orbits of Hamiltonian systems. After reviewing background and some previous results, I will explain a formula which relates symplectic capacity of (fiberwise) convex domains to loop space homology, and discuss some applications and questions.

SUMMARY: Asymptotically (locally) hyperbolic spaces are certain non-compact complete Riemannian manifolds with the property that the name suggests. A remarkable feature of such a space is that its boundary at infinity is naturally equipped with a conformal structure (not necessarily locally flat); it is of interest how this conformal structure affects the analytic property of the space. We can also consider asymptotically complex hyperbolic spaces, whose boundary carries a CR structure (Cauchy–Riemann structure). These two instances are actually expected to continue to (probably) an infinite number of fruitful correspondences involving various types of “parabolic geometries.”

In this talk, I will try to convey general ideas about geometric analysis on asymptotically hyperbolic and complex hyperbolic spaces through discussing aspects of the “Einstein filling problem”—that is, the problem of finding such a space satisfying the Einstein equation with given conformal or CR structure on the boundary. Examples indicating the (fun and) subtleties of the problem will be presented. Of central theoretical importance is the Fredholm theorem for geometric linear elliptic differential operators due to O. Biquard, J. Lee, and J. Roth, from which some decent perturbation theorems for Einstein metrics follow. I will also propose a new approach in the complex case, which is to strengthen the filling structure by attaching compatible almost complex structures to Einstein metrics.

SUMMARY: A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In my talk I will introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.

SUMMARY: In a previous MSJ meeting we gave an explicit description of maximal antipodal sets of Riemannian symmetric spaces related to the exceptional compact Lie group \(G_2\). Using this description we explain close relation between the algebraic structure of the octonions and the Fano plane.

SUMMARY: We consider local isometric embeddings of 3-dimensional warped product metrics into \(\mathbb {R}^4\). By calculating the curvature and its covariant derivative of this metric, we first obtain a necesssary condition to admit isometric embeddings of this space into \(\mathbb {R}^4\). Conversely, for a generic case, we show that this condition is sufficient to ensure the existence of local isometric embeddings into \(\mathbb {R}^4\). By solving an ordinary differential equation, we explicitly determine the form of warped product metric that can be locally isometrically embedded into \(\mathbb {R}^4\).

SUMMARY: It is well known that Frenet frame of a spacial curve is in Euclidian Space, but L. R. Bishop propsed that other frame is in the Euclidian Space. It is called Bishop frame. This frame is very useful for describing some particular curve. In 3-dimenssional Euclidean Space, Bishop considered 3 types of coefficient matrixs, one of them is the same as Frenet frame by changing basis. So we consider 4 types of frames in 4-dimentional Euclidian Space and we consider some kind of degree of strengths of frames by coefficient matrixs.

SUMMARY: We introduce the classification theorem of real hypersurfaces in a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space) from the viewpoint of the parallelism of a certain tensor.

SUMMARY: Existence of the positive scalar curvature (psc) metric has been an important topic in differential topology of higher dimensional manifolds, particularly in the presence of fundamental groups. The Rosenberg index of a closed spin manifold is a generalization of the Atiyah–Singer index, which is a topological obstruction of the existence of a psc metric. In 2014 Hanke–Pape–Schick shows that the Rosenberg index of a codimension 2 submanifold \(N\) obstructs the existence of a psc metric on \(M\) by using the coarse geometry of covering spaces. Here we give a different understanding of the argument of Hanke–Pape–Schick in order to strengthen their result; we show that the nonvanishing of the Rosenberg index of \(N\) implies that of \(M\).

SUMMARY: It is known that the Lichnerowicz estimate for the first eigenvalue of the Laplacian acting on functions is improved when the Riemannian manifold has a non-trivial parallel differential form. In this talk, we consider the situation such that the Riemannian manifold has a non-trivial almost parallel differential form, and show that the Lichnerowicz estimate is improved then. Moreover, we give a pinching result about the almost equality case of the estimate.

SUMMARY: Symplectic-Haantjes manifolds are constructed for several Hamiltonian systems following Tempesta–Tondo, which yields the complete integrability of systems. In this talk, we consider an approach to integrable systems by constructing concrete examples of Symplectic-Haantjes manifolds.

SUMMARY: In this talk, we discuss the Hermite–Liouville structures on Hopf surface. We construct a one-parameter family of the structures by deforming its metric and its orthonormal frame, and find the first integrals on their cotangent bundles of their geodesic flows. We also see the complete integrability of their geodesic flows by virtue of the structures and the first integrals. The argument in this talk is in relation to the previous talks given by the speaker at the MSJ Spring Meeting 2019 and at the MSJ Spring Meeting 2018.

SUMMARY: In studies of the AdS/CFT correspondence, Graham and Witten have introduced the area renormalization. By using this procedure, we can define an invariant for immersions from an even-dimensional closed manifold to a conformal manifold, called the Graham–Witten energy. In this talk, we will discuss the variation of this invariant under deformations of immersions.

SUMMARY: We construct a family of invariant Riemannian metrics and affine connections on the space of positive probability measures on a finite state space under \(\phi \)-Markov embeddings when the space of probability measures is embedding into the Euclidean space with distortion \(\phi \).

SUMMARY: We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying \(\mathrm {Ric}_{\infty } \ge K>0\). Assuming that equality holds, we show that the \(1\)-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng–Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems.

SUMMARY: It is known that Monge–Ampère systems is a geometric formalization of Monge–Ampère equations using the theory of exterior differential systems. In this talk, we give a generalization of Monge–Ampère systems, Monge–Ampère equations and a relationship between such systems and equations.

SUMMARY: I will concentrate on the Monge mass transportation problem with distance cost. The existence of optimal transport maps in non-branching metric spaces with some lower curvature bounds has been well studied. On the other hand, it does not seem that the study of the Monge problem in the branching case is adequate. In this talk, I will show the existence of an optimal transport map for some projective metrics on a convex domain in Euclidean space. The main result is applicable to metric spaces, admitting branching geodesics, such as Hilbert geometries and bounded domains in a normed space.

SUMMARY: Based on Getzler’s rescaling transformation, we obtain a formula for the heat kernel coefficients of the Dirac Laplacian on a spin manifold. One can compute them explicitly up to an arbitrarily high order by using only a basic knowledge of calculus added to the formula.

SUMMARY: We defined harmonic manifolds of hypergeometric type, which is a class of harmonic manifolds including rank-one symmetric space of non-compact type and Damek–Ricci spaces. In this talk, we present that the volume entropy \(Q\) of an \(n\)-dimensional harmonic Hadamard manifold \((X, g)\) of hypergeometric type, normalized as \(\mathrm {Ric}_g=-(n-1)g\), satisfies the inequality \(\frac {2\sqrt {2}(n-1)}{3}\le Q\le n-1\), and the equality \(Q=n-1\) holds if and only if \((X,g)\) is the real hyperbolic space of constant sectional curvature \(-1\).

SUMMARY: We show a method to construct a Lagrangian mean curvature flow from a given special Lagrangian submanifold in a Calabi–Yau manifold by generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. We construct some examples in \(\mathbb {C}^n\) by our method.

SUMMARY: It is known that real special linear group \(SL(2,{\mathbb R})\) is embedded into the three-dimensional sphere. By the stereographic projection, every matrix in \(SL(2,{\mathbb R})\) is realized as a point in the three-dimensional Euclidean space \({\mathbb R}^3\). In this talk, we propose another three-dimensional model of \(SL(2,{\mathbb R})\). With this model, we can visualize \(SL(2,{\mathbb Z})\) as a pattern of points on cubic lattice in \({\mathbb R}^3\). In this model, the set of matrices with the fixed value of trace forms a quadratic surface (hyperboloid of two sheets, double cone, or hyperboloid of one sheet) depending on the value of trace. Hyperbolic paraboloid also comes out as the surface of the fixed value of element. With these familiar surfaces, we can analyze the pattern of \(SL(2,{\mathbb Z})\).

SUMMARY: We study the irreducible decomposition of the unitary representations of the Heisenberg group. We apply the Poisson sigma model for this purpose. By using the orbit of the parabolic Toda lattice in the target space, we can define the polarization on X=U/R, where U is the Heisenberg group and R is its center. The symplectic structure of X is defined by the Poisson relations on orbit of the Toda lattice in the target space. The central charges (s) and pull back of the target space (x) make the moduli space of the symplectic structures of X. We consider the quantization of the action of U on the moduli space.

SUMMARY: By the SYZ construction, a mirror pair \((X,\check {X})\) of a complex torus \(X\) and a mirror partner \(\check {X}\) of the complex torus \(X\) is described as the special Lagrangian torus fibrations \(X \to B\) and \(\check {X} \to B\) on the same base space \(B\). Then, by the SYZ transform, we can construct a simple projectively flat bundle on \(X\) from each affine Lagrangian multi section of \(\check {X} \to B\) with a unitary local system along it. However, there are non-unique choices of transition functions of it, and this fact actually causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this talk, by solving this problem, we prove that there exists a bijection between the set of the isomorphism classes of their objects.

SUMMARY: Many examples are known where the initial problem of differential equation is solved by exponential matrix or its matrix elements. Some of those solutions have the geometrical interpretations, e.g., Toda lattice, Calogero system and other integral systems. As such an example, we see the geodesic equation of a statistical manifold that is homogeneous but is not symmetry. We give its solution an interpretation by the adjoint representation of semisimple Lie algebras.

SUMMARY: Lie groups with left-invariant Riemannian metrics have provided many interesting and nice examples of Riemannian manifolds, such as Einstein or Ricci soliton. For a given Lie group, the existence and non-existence problems of some nice left-invariant Riemannian metrics are interesting. In order to attack these problems, we focus on Lie groups whose moduli spaces of left-invariant Riemannian metrics are small. In this talk, we give a classification of almost abelian Lie groups whose moduli spaces of left invariant Riemannian metrics are one-dimensional.

SUMMARY: The classification problem of the homogeneous spaces which admit compact Clifford–Klein forms (also called compact quotients) is one of the important open problems. We consider this problem for a class of indecomposable and reducible pseudo-Riemannian symmetric space of solvable type. In previous research by I. Kath–M. Olbrich, this problem was attacked by using the property of solvable Lie group. In this talk, we show a necessary condition for the existence of compact Clifford–Klein forms for the class by another method. This method using relative cohomology was introduced by T. Kobayashi and K. Ono and was developed by Y. Morita.

SUMMARY: In 2017, Sunada proved theorems on distribution for the set \(\Gamma _1\) of primitive lattice points related to a problem in Gauss’ Mathematisches Tagebuch (diary) and an arithmetic discrete set \(\Gamma _2\) related to primitive Pythagorean triples (PPTs). In addition, he gave an alternative proof to Lehmer’s asymptotic theorem for PPTs, using a certain “summation formula”. We observe that a “summation formula” holds also for another arithmetic discrete set \(\Gamma _3\) related to primitive Eisenstein triples (PETs). This allows to obtain an asymptotic behavior of PETs.

SUMMARY: In the joint work with Oguni, we introduced a new class of “non-positively curved” metric spaces in coarse geometry. Recently Huang and Osajda showed that Artin groups of type FC and weak Garside groups of finite type act geometrically on Helly graphs. Their result gives us many examples of groups acting geometrically on coarsely convex spaces.

SUMMARY: In this talk, we develop the theories of symmetric triads with multiplicities and double Satake diagrams. We give a one-to-one correspondence between compact symmetric triads and double Satake diagrams. As its applications, we obtain an alternative proof for Matsuki’s classification theorem for compact symmetric triads in terms of double Satake diagrams. Further, we give a natural correspondence between commutable compact symmetric triads and symmetric triads with multiplicities.

SUMMARY: The notion of coupled Kähler–Einstein metrics was introduced recently by Hultgren–W. Nyström. In this talk, we discuss deformation of coupled Kähler–Einstein metrics on Fano manifolds. In particular, we obtain a necessary and sufficient condition for a coupled Kähler–Einstein metric to be deformed to a coupled Kähler–Einstein metric for another close decomposition for anti-caonnical class of Fano manifolds admitting non-trivial holomorphic vector fields. This generalizes a result of Hultgren–W. Nyström.

SUMMARY: We introduce a Kähler-like almost Hermitian metric and an almost balanced metric. We prove that on a Kähler-like almost Hermitian manifold, we have an identity between the first derivative of the torsion \((1,0)\)-tensor and the Nijenhuis tensor. By applying the identity, then we figure out what the equivalent condition of being almost balanced on a compact Kähler-like almost Hermitian manifold is. Moreover, we prove that on a compact Kähler-like almost Hermitian manifold \((M^{2n},J,g)\), if it admits a positive \(\partial \bar {\partial }\)-closed \((n-2,n-2)\)-form, then \(g\) is a quasi-Kähler metric.

SUMMARY: We clarify the relation between Calabi’s extremal Kähler metrics and Mabuchi’s Kähler–Einstein metrics on toric Fano manifolds by comparing the corresponding stabilities.

SUMMARY: The Riemann–Roch theorem for tropical curves was shown by Gathmann–Kerber and Mikhalkin–Zharkov in 2008. It is a very interesting problem to generalize the tropical Riemann–Roch theorem to higher dimensions, while there are few results for this problem. A main obstacle to higher dimensional generalization is to define the Euler characteristic of a tropical line bundle since the higher cohomology of line bundles cannot be defined as ordinary way. In this talk, we show the Riemann–Roch inequality for tropical abelian surfaces and more results by studying global sections of line bundles over tropical tori, called tropical theta functions.

SUMMARY: Roughly speaking, a manifold is said to be formal if the real homotopy type is determined by its cohomology. A formal manifold has the trivial Massey product, which gives a topological obstruction for a manifold \(M\) to be formal.

The notion of the formality is defined for differential graded algebras (DGAs). We study it for special DGAs called Poincaré DGAs of Hodge type. Applying this to a manifold, we obtain some topological obstructions for a manifold to admit a geometric structure.

SUMMARY: We introduce the new notion of a homogeneous pair for a pseudo-Riemannian metric \(g\) and a positive function \(f\) on a manifold \(M\). We consider the conformal transformations of \(g\) using \(f\) and study the geometric structures such as the curvature, geodesics and the metric completion (if \(g\) is positive definite).

We have many examples that admit this structure. In particular, many moduli spaces of geometric structures admit this structure. We provide the unified method for the study of geometric structures of these manifolds.

SUMMARY: I will talk about our recent result on algebraic description of a wide class of twistor spaces associated to anti-self-dual metrics on compact 4-manifolds. Each of these twistor spaces is birational to the total space of a del Pezzo fibration over CP1, and may be described by a single quartic polynomial of a particular form. Generic fibers of the fibration are (possibly singular) del Pezzo surfaces of degree two.