2019年度年会(於:東京工業大学)

SUMMARY: Spin geometry deals with the Dirac operator and spinors on spin manifolds. One of the famous theorems is that there exists no non-trivial harmonic spinor on a positive scalar curvature manifold because of Lichnerowicz’s formula. What happen for the spin \(3/2\) case? As stated in the physics literature, the Rarita–Schwinger operator on spin \(3/2\) fields is an analog of the Dirac operator. If a spin \(3/2\) field is in the kernel of the Rarita–Schwinger operator, we call it a Rarita–Schwinger field. In contrast to spin \(1/2\) case, positive scalar curvature is not the condition to rule out the existence of RS fields. In fact, we can find examples of compact Einstein manifolds with/without RS fields, where the key is to use a variety of Weitzenböck formulas. For instance, we have a complete classification of quaternionic-Kähler manifolds and symmetric spaces admitting RS fields. In this talk, I will present recent results by a joint research with U. Semmelmann for RS fields and related Weitzenböck formulas.

SUMMARY: It is known that any compact Riemann surface admits a unique constant Gaussian curvature (Hermitian) metrics. Extending it to higher dimensional complex varieties, there is a notion of Kähler–Einstein metrics which is canonical (unique), characterized by constancy of Ricci curvature. The sign of Ricci curvature crucially controls the geometric properties, especially when we take limit spaces.

In our studies done while ago, we worked on relations between such metrics and birational algebraic geometry, and then algebra-geometric compactification of moduli space of Fano varieties, positive Ricci curvature case.

The focus of this talk will be, among others, on the case Ricci curvature is zero, so-called “Calabi–Yau metrics” or Ricci-flat Kähler metrics. Our recent work with Yoshiki Oshima (arXiv:1810.07685) provides a moduli-theoretic framework for the collapsing of Ricci-flat Kähler metrics by certain explicit compactifications of classical moduli varieties. The speaker originally called the obtained compactification “tropical geometric compactification” and the joint work largely develops the theory.

On the way, what we observe in various forms repeatedly are two general deep natures of Kähler–Einstein metrics, its “algebraicity” (or “algebro-geometricity”) and “minimality”.

SUMMARY: We say that an effective action of a compact torus \(G\) on a connected smooth manifold \(M\) is maximal if there exists a point \(x \in M\) such that \(\dim G+\dim G_x = \dim M\). We give a complete classification of compact connected complex manifolds with maximal torus actions, in terms of combinatorial objects, which are triples \((\Delta , \mathfrak {h}, G)\) of nonsingular complete fan \(\Delta \) in \(\mathfrak {g}\), complex vector subspace \(\mathfrak {h}\) of \(\mathfrak {g}^\mathbb {C}\) and compact torus \(G\) satisfying certain conditions.

On the other hand, a compact connected complex manifold equipped with a compact torus action has a holomorphic foliation coming from the torus action. We discuss a classification of compact connected complex manifolds with maximal torus actions up to transverse equivalence. If time permits, we also discuss the basic cohomology and basic Dolbeault cohomology of such manifolds.

SUMMARY: We investigated oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the 3-dimensional anti-de Sitter space. It is known that such Lorentzian surfaces are either a B-scroll or a complex circle. We determined the type numbers of the pseudo-hyperbolic Gauss maps of a B-scroll and a complex circle. Also, we investigated the behavior of the type numbers of the pseudo-hyperbolic Gauss maps along their parallel families.

SUMMARY: Statistical manifolds are manifolds endowed with a torsion-free affine connection and a Riemannian metric. A statistical manifold is said to be a Hessian manifold if its affine connection is flat. A diffeomorphism between statistical manifolds is said to be statistical if it preserves statistical structures. Our purpose is to find conditions that guarantee an extension of a given linear isomorphism between given tangent spaces to a local statistical diffeomorphism. We explain that a statistical structure is locally characterized by its Riemannian curvature tensor and difference tensor. In addition, we also show that a Hessian structure is locally determined by its Hessian curvature tensor and difference tensor.

SUMMARY: Calabi’s Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz–Minkowski space which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space and maximal surfaces in Lorentz–Minkowski space, we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph which does not admit time-like points (namely, the graph consists of only space-like and light-like points) is a plane.

SUMMARY: A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this talk, we exhibit several results on the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss–Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

SUMMARY: We construct constant negative Gaussian curvature tori with one family of planar curvature lines in Euclidean 3-space. The singularities of these tori are studied.

SUMMARY: In this talk, we present the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.

SUMMARY: Let \(M\) be an orientable finitely connected and geodesically convex Finsler 2-manifold with genus \(g \ge 1\). We assume that some closed geodesics are reversible. However, the 2-manifold \(M\) does not need to be complete and without boundary. We prove that for any number \(\varepsilon > 0\) and for any points \(p, q \in M\) there exists a number \(R > 0\) such that any geodesic circle with center \(p\) and radius \(t\) meets the \(\varepsilon \)-ball with center \(q\) if \(t > R\).

SUMMARY: In this talk, we discuss the Elliptically deformed Hopf surfaces with hermitian metrics, and construct Hermite–Liouville structures on them and find the first integrals on their cotangent bundles of their geodesic flows. Also, we see the complete integrability of their geodesic flows by virtue of the structures and the first integrals. The argument in this talk is a continuation of the previous talk given by the speaker at the MSJ Spring Meeting 2018.

SUMMARY: This talk is based on a joint work with Tudor Ratiu (Shanghai Jiao Tong University). The stability of the isolated equilibria is considered for Euler equation of the Mishchenko–Fomenko geodesic flow on an arbitrary real semi-simple Lie group, by using the results of Bolsinov and Oshemkov for bi-Hamiltonian systems. It is shown that the type of an isolated equilibrium on a generic orbit can be characterized by the respective numbers of the real, purely imaginary, and complex roots.

SUMMARY: We show that Hermitian–Einstein metrics can be constructed locally by a map of (anti-) self-dual bifurcations on Euclidean \(R^4\) to symmetric two-tensors introduced in “Gravitational instantons from gauge theory,” H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence applies not only to a commutative space, but also to a non-commutative space. We choose U(1) instantons on a noncommutative \(C^2\) as a self-dual form, from which we derive a family of Hermitian–Einstein metrics. We also discuss the condition when the metric becomes Kaehler.

SUMMARY: We show 10/8-type inequalities for some end-periodic 4-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed 4-manifolds which do not admit positive scalar curvature metrics.

SUMMARY: We have studied asymptotic diameter growth of the instanton moduli spaces of the four-sphere, which partly solved Donaldson’s conjecture.

SUMMARY: First of all, we introduce the relation between vector bundles over a manifold and maps from the manifold into Grassmannian manifolds. And then we classify harmonic maps from complex projective line into the complex Grassmannian manifolds of two-planes which have certain conditions.

SUMMARY: In this talk, we construct a one-parameter family of equivariant holomorphic embedding from complex projective space into the complex Grassmannian.

SUMMARY: We considered the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We proved that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition and its image by the orbit map of the Hilbert Lie group action is included by the geodesic ball of some radius, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. As an application of this result to the gauge theory, we derived a result for the behavoiur of the holonomies (along a fixed loop) of connections belonging to some based gauge-invariant hypersurface in the space of connections on the principal bundle having a compact semi-simple Lie group as the structure group along a natural flow starting from the hypersurface.

SUMMARY: We establish a new compactness theorem for complete Riemannian manifolds via \(m\)-Bakry–Émery Ricci curvature with positive \(m\). Our result generalizes the Myers-type theorem via \(m\)-Bakry–Émery Ricci curvature by M. Limoncu (2010) and may be compared with Ambrose- and Cheeger–Gromov–Taylor-type theorems via \(m\)-Bakry–Émery Ricci curvature by the author (2016).

SUMMARY: We establish some new compactness theorems for transverse Ricci solitons on complete Sasaki manifolds. Our results are natural generalizations of the Myers-type theorem by M. Fernández-López and E. García-Río (2008) and M. Limoncu (2010), and the Cheeger–Gromov–Taylor-type theorem by the author (2016).

SUMMARY: As metrics on the set of all metric measure spaces, there are the box distance and the observable distance introduced by Gromov. The topology induced by the observable distance is called the concentration topology and is weaker than one induced by the box distance. In this talk, I address a question which asks a convergence of \(l_p\)-product spaces for two convergent sequences of metric measure spaces. For the box topology, this problem is easy. However, for the concentration topology, this problem is harder. The main result gives the answer of this problem for the concentration topology.

SUMMARY: This talk will present the speaker’s recent results on a “canonical” Laplacian on some round Sierpiński carpets (RSCs), i.e., subsets of \(\mathbb {C}\cup \{\infty \}\) homeomorphic to the standard Sierpiński carpet with complement consisting of disjoint open disks. On the Apollonian gasket, Teplyaev (2004) had constructed a canonical Dirichlet form as one with respect to which the coordinate functions are harmonic, and the speaker later proved its uniqueness and an explicit expression in terms of the circle packing structure of the gasket. This last expression of the Dirichlet form makes sense on general circle packing fractals, including RSCs, and defines a “canonical” Laplacian on such fractals. Moreover, with the knowledge of some combinatorial structure of the fractal it is also possible to prove Weyl’s eigenvalue asymptotics for this Laplacian.

SUMMARY: In this talk I consider the continuity of the eigenvalues of the connection Laplacian of \(G\)-connections on vector bundles over Riemannian manifolds. To show it, I introduce the notion of the asymptotically \(G\)-equivariant measured Gromov–Hausdorff topology on the space of metric measure spaces with isometric \(G\)-actions, and apply it to the total spaces of principal \(G\)-bundles equipped with \(G\)-connections over Riemannian manifolds.

SUMMARY: We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound.

SUMMARY: We investigate the relationship between geometric and analytic indices for quotients of the Cayley graph of the free group \({\rm Cay}(F_n)\). Our main result, which generalises Grigorchuk’s cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on \(G \backslash {\rm Cay}(F_n)\) to the Poincaré exponent of \(G\). Our main tool is the Patterson–Sullivan theory for metric trees.

SUMMARY: We prove that if \(n\geq 5\), there is no area formula of the general hyperbolic and spherical cyclic \(n\)-gon written in terms of arithmetic operations and \(k\)-th roots of its side lengths.

SUMMARY: It is known that the centroid, the incenter and the Chebyshev center of a body in Euclidean space are obtained as critical points of the Riesz potential of the body. Applying this fact to a triangle in Euclidean plane, we review and generalize the classical theorem that if at least two of the centroid, the incenter and the circumcenter of a triangle coincide, then the triangle has to be regular.

SUMMARY: In order to construct hyperbolic 4-manifolds, Kolpakov and Slavich introduced the coloring technique of right-angled 4-polytopes. In this talk, we consider the hyperbolic Coxeter 4-polytope defined by a Napier cycle and construct hyperbolic 4-manifolds by using the coloring technique.

SUMMARY: We propose a new formulation of quantized algebra by using category theory. There are several ways of quantization of algebra, for example, deformation quantization, matrix regularization and so on. For the unified description of them, we define quantization of an algebra as a functor. A sequence of some categories of algebras is a sequence of corresponding quantized algebras, and the limit of them is a classical algebra. We discuss deformation quantization and matrix regularization closely as examples.

SUMMARY: We define bow varieties for the symplectic group as quotients of some appropriate vector spaces by products of general linear and orthogonal groups. If we impose the balanced condition, we have equidimensionality of fibers of their factorization maps, and that these varieties are normal. We expect these two properties can be used to obtain a relationship between these varieties and Coulomb branches of quiver gauge theories of affine type C.

SUMMARY: We observe that a system of irreducible, fiber-linear, first class constraints on \(T^*M\) is equivalent to the definition of a foliation Lie algebroid \(E\) over \(M\). The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description \((E[1],Q)\) of the Lie algebroid to its cotangent bundle \(T^*E[1]\). Adding a Hamiltonian to the system corresponds to a metric \(g\) on \(M\). Consistency introduces a connection \(\nabla \) on \(E\) and one obtains the compatibility of \(g\) with \((E,\rho ,\nabla )\). This leads a geometric construction of a BFV and BV-AKSZ formalism.

SUMMARY: In this talk, I introduce a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a closed oriented 3-manifold, and show the constrained Hamiltonian system of the function produces \(\mathrm {SO}(3)\)-invariant \(G_2\)-manifolds.

SUMMARY: We introduce two parabolic flows which preserve the almost pluriclosedness and the almost balancedness respectively in the almost Hermitian geometry. The first one is called an almost pluriclosed flow. We show that the flow has a unique short-time solution and also show that this flow coincides the palabolic flow called an almost Hermitian curvature flow. The second one is a parabolic flow of almost Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature, which is called a scalar Calabi-type flow. We show that the flow has a unique short-time solution and also show a stability result when the background metric is quasi-Kähler with constant scalar curvature.

SUMMARY: Let \(\Delta \) be a smooth reflexive polytope in dimension 3 and \(f\) be a tropical polynomial whose Newton polytope is the polar dual of \(\Delta \). One can construct a \(2\)-sphere equipped with an integral affine structure with singularities by contracting the tropical K3 hypersurface defined by \(f\). We write the complement of the singularity as \(i \colon B_0 \hookrightarrow B\), and the local system of integral tangent vectors on \(B_0\) as \(T\). In the talk, we will give a primitive embedding of the Picard group \(\mathrm {Pic} X\) of the toric variety \(X\) associated with the normal fan of \(\Delta \) into \(H^1(B, i_\ast T)\), and compute the radiance obstruction of \(B\), which sits in the image of \(\mathrm {Pic} X\). We will also discuss the relation with the asymptotic behavior of the period map of complex K3 hypersurfaces.

SUMMARY: In this talk, I give an \(\varepsilon \)-regularity theorem for line bundle mean curvature flows. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang–Mills metrics on a given Kähler manifold and recently defined by Jacob and Yau. To establish the theorem, I will introduce a scale invariant monotone quantity and as the critical point of this quantity self-shrinker solutions of the line bundle mean curvature flow are defined. The Liouville type theorem for self-shrinkers will be also given and it plays an important role in the proof of the \(\varepsilon \)-regularity theorem.

SUMMARY: We show a method to construct a special Lagrangian submanifold \(L^\prime \) from a given special Lagrangian submanifold \(L\) in a Calabi–Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in non-flat Calabi–Yau manifolds \(\mathrm {T}^\ast S^n\) which equipped with the Stenzel metrics.

SUMMARY: We investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space \(\mathbb {C}H^n\). We consider a standard Hamiltonian \(T^n\)-action on \(\mathbb {C}H^n\), and show that every Lagrangian \(T^n\)-orbits in \(\mathbb {C}H^n\) is H-stable when \(n\leq 2\) and there exist infinitely many H-unstable \(T^n\)-orbits when \(n\geq 3\). On the other hand, we prove a monotone \(T^n\)-orbit in \(\mathbb {C}H^n\) is H-stable and rigid for any \(n\). Moreover, we see almost all Lagrangian \(T^n\)-orbits in \(\mathbb {C}H^n\) are not Hamiltonian volume minimizing when \(n\geq 3\) as well as the case of \(\mathbb {C}^n\) and \(\mathbb {C}P^n\).

SUMMARY: LCK manifold is said to be a Vaisman manifold if Lee form is parallel with respect to Levi–Civita connection. In this talk, we prove that if a solvmanifold such that the commutator of the solvable Lie group is abelian has a Vaisman structure, then it is Kodaira–Thurston manifold. As corollary, a solvmanifold such that the solvable Lie group is meta-abelian has no Vaisman structures.