2018年度秋季総合分科会(於:岡山大学)

SUMMARY: If a discrete subgroup \(\Gamma \) of a Lie group \(G\) acts properly and freely on a homogeneous space \(G/H\), the quotient space \(\Gamma \backslash G/H\) becomes a manifold locally modelled on \(G/H\), and is called a Clifford–Klein form. Since the initial work by T. Kobayashi (1989), the global geometry and topology of Clifford–Klein forms in the non-Riemannian case (i.e. the case when \(H\) is noncompact) has been studied by various methods. In this talk, I will explain some necessary conditions for the existence of compact Clifford–Klein forms obtained by comparing relative Lie algebra cohomology and de Rham cohomology. I will also discuss their reinterpretations in terms of invariant polynomials and Sullivan algebras.

SUMMARY: In this talk we will discuss the recent developments on the study of metric measure spaces with Ricci bounds from below and applications to Riemannian geometry. In particular it is explained how to construct nontrivial geometric/analytic quantities which are continuous with respect to measured Gromov–Hausdorff convergence.

SUMMARY: A Fano manifold admits a Kähler–Einstein metric if and only if it is K-polystable. This theorem was established by Chen–Donaldson–Sun around 2012. Such a standard metric is characterized as the critical point of the canonical energy functional and in fact the existence is equivalent to the proper growth condition of the energy, which implies that the modulus of stability can be taken uniformly in test configurations of the manifold. The idea is also for the purpose of attacking general constant scalar curvature metric problem and even in the Kähler–Einstein case it leads Berman–Boucksom–Jonsson to a new simple proof of the existence. More recently the unstable case attracts people’s interest and we expect the parabolic version of the theorem of Chen–Donaldson–Sun. Namely, the gradient flow of the energy functional should produce a unique test configuration which optimally destabilizes the manifold. In collaboration with T. Collins and R. Takahashi we showed the long-time existence of the flow. The relevant soliton metric and stability will be also discussed.

SUMMARY: In this talk I will talk about developments of index theory of Dirac-type operator on symplectic manifolds. In particular I will focus on localization phenomena of index and localization technique using perturbation by Dirac-type operator along fibers. I am planning to talk about the following contents.
1. A motivation: geometric quantization
2. Localization formula of index: perturbation by Dirac-type operator along fibers
3. Applications
4. Further developments

SUMMARY: In this talk, we will introduce new space results from the viewpoint of geometry from the book manuscript of division by zero calculus in figures —Our new space since Euclid— (Draft).

SUMMARY: From the viewpoint of the division by zero (0/0=1/0=z/0=0) and the division by zero calculus, we will show interesting applications to Wasan geometry that show unexpected new discovery for some extreme cases. As a typical example, we will show a typical result for the old Japanese geometry, clearly.

SUMMARY: We consider the diffusion system on an evolving double bubble from an energetic point of view. We employ an energetic variational approach to make a mathematical model of the diffusion system on the double bubble. Moreover, we study the boundary conditions for our system to investigate both conservation and energy laws of the system.

SUMMARY: In this talk, we investigate surfaces in singular semi-Euclidean space \(\mathbb {R}^{0,2,1}\) endowed with a degenerate metric. We define \(d\)-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we show that \(d\)-minimal surfaces in \(\mathbb {R}^{0,2,1}\) and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space are in one-to-one correspondence.

SUMMARY: The classical Bianchi–Bäcklund transformation for constant mean curvature surfaces in Euclidean 3-space has been studied by many researchers. In this talk, we introduce the method to construct positon-like solution of elliptic sinh-Gordon equation via successive Bianchi–Bäcklund transformations with a single spectral parameter. We also show the recipe of the corresponding constant mean curvature surfaces of positon-like solutions.

SUMMARY: Minimal surfaces with planar curvature lines and minimal surfaces which are also affine minimal in the Euclidean space have been studied since the late 19th century and early 20th century. In this talk, we reveal that timelike minimal surfaces in the Minkowski space which are also affine minimal consist of timelike minimal surfaces with planar curvature lines and their conjugates, and they are generated by null curves with constant lightlike curvature. We also give a classification, including results of deformations and singularities, of such surfaces.

SUMMARY: We compute principal curvatures of homogeneous hypersurfaces in a Grassmann manifold \(\widetilde {{\rm Gr}}_{3}({\rm Im}\mathbb {O})\) by the \(G_2\)-action. As applications, we find an orbit which is an austere submanifold and orbits which are proper biharmonic homogeneous hypersurfaces. We also show that an orbit is a weakly reflective submanifold.

SUMMARY: We consider the Grassmann manifold of associative subspaces in the space of imaginary octonions, which we call the associative Grassmann manifold. It is known that the associative Grassmann manifold is an eight-dimensional compact symmetric quaternionic Kähler manifold. We construct interesting examples of four-dimensional complex submanifolds of the associative Grassmann manifold.

SUMMARY: Real special linear group \(SL(2,{\mathbb R})\) is embedded into the three-dimensional sphere. We can see the three-dimensional sphere by the stereographic projection. Through this visualization, every matrix in \(SL(2,{\mathbb R})\) is realized as a point in the three-dimensional Euclidean space. The set of symmetric matrices is on a Euclidean plane, and on this plane \(SL(2, {\mathbb Z})\) makes a certain hyperbolic pattern.

SUMMARY: Let \(G\) be a reductive group, \(P\) be its parabolic subgroup, and \(H\) be a closed subgroup of \(G\). There are several studies on the orbit decomposition of the flag variety \(G/P\) by the \(H\)-action, and these studies are expected to play an important role in various problems such as branching problem of \(G\) with respect to \(H\). These studies were mostly based on the cases where \(H\) has only finitely many orbits on the flag variety. We focus on explicit descriptions of the orbit decomposition of a multiple flag variety \((G\times G\times \cdots \times G)/(P_1\times P_2\times \cdots \times P_m)\) by the diagonal action of \(G\) with infinitely many orbits.

SUMMARY: I will talk about the existence problem of compact Clifford–Klein forms for pseudo Riemannian symmetric spaces. I will show that in the class of indecomposable pseudo Riemannian symmetric space with signature \((2,2)\), only one space up to isomorphism admits compact Clifford–Klein froms. To elucidate that, I have divided this problem into two cases – the first case in which the symmetric space \(G/H\) is completely solvable type and the second case is not. In the first case, for a compact Clifford–Klein form \(\Gamma \backslash G/H\), there exists a connected and closed subgroup \(L\) which includes \(\Gamma \) cocompactly. Therefore, we can give the existence condition by using Lie algebra. In the second case, there is not necessarily such subgroup, I define a non-conneted but ‘almost’ connected subgroup, and simplify the existence problem of compact Clifford–Klein forms into that of the subgroup.

SUMMARY: Cohomogeneity one actions of connected Lie groups on irreducible symmetric spaces of noncompact type have been classified into three types. In this talk, we study cohomogeneity one actions of disconnected Lie groups, and extend the above classification. We also study some relationships between the conditions for existence of such actions and geometry of their orbits.

SUMMARY: In this talk, we give a necessarily and sufficient condition for orbits of linear isotropy representations of Riemannian symmetric spaces are biharmonic submanifords in hyperspheres. In particular, we obtain examples of biharmonic submanifolds in hyperspheres whose co-dimension is greater than one.

SUMMARY: In this talk, we give the definition of generalized \(s\)-manifolds as a generalization of symmetric spaces. Moreover, for a generalized \(s\)-manifolds, we introduce the notions of polars and antipodal sets, and define the antipodal number as the supremum of the cardinalities of antipodal sets.

We give \(s\)-structures on Flag manifolds, and determine maximal antipodal sets and the antipodal number.

SUMMARY: In previous MSJ meetings we gave the classification of maximal antipodal subgroups of the quotient groups of classical compact Lie groups. Using this classification we show the classification of maximal antipodal sets of classical compact symmetric spaces.

SUMMARY: We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that symmetry-breaking bifurcation does not occur except at the round metric.

SUMMARY: For a closed, connected direct product Riemannian manifold \((M, g)=(M_1\times \cdots \times M_l, g_1\oplus \cdots \oplus g_l)\), we define its multiconformal class \([\![g]\!]\) as the totality \(\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}\) of all metrics obtained from multiplying each \(g_i\) by a function \(f_i^2>0\) on the total space \(M\). A multiconformal class \([\![g]\!]\) contains not only all warped product type deformations of \(g\) but also the whole conformal class \([\tilde {g}]\) of every \(\tilde {g}\in [\![g]\!]\). We prove that \([\![g]\!]\) carries a metric of positive scalar curvature if and only if the conformal class of some factor \((M_i, g_i)\) does, under the technical assumption \(\dim M_i\ge 2\). We also show that, even in the case where every factor \((M_i, g_i)\) has positive scalar curvature, \([\![g]\!]\) carries a metric of scalar curvature constantly equal to \(-1\) and with arbitrarily large volume, provided \(l\ge 2\) and \(\dim M\ge 3\). In this case, we observe that such negative scalar curvature metrics within \([\![ g]\!]\) cannot be of any warped product type, provided \(l=2\).

SUMMARY: We consider the quotient space of the Heisenberg group by its center. The Heisenberg group it self is the principal bundle of the quotient space. Thus one can define the line bundle over the quotient space by using the character of the center. The symplectic structure of the quotient space is defined by the curvature of the line bundle whose characterlistic class is integral. We consider the the symplectic structures of the quotient space of non integral classes and the relationship to the Stone–von Nuemann theorem.

SUMMARY: We talk about the stability of the Riemannian curvature dimension condition introduced by Ambrosio–Gigli–Savaré under the concentration of metric measure spaces introduced by Gromov. This is an analogue of the result of Funano–Shioya for the curvature dimension condition of Lott–Villani and Sturm. These conditions are synthetic lower Ricci curvature bound for metric measure spaces.

SUMMARY: We investigate the combinatorial Ricci flow on a torus when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.

SUMMARY: For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David–Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this talk, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces exactly has continuum cardinality.

SUMMARY: A triplet \((X , d , T)\) is called an integral current space if \((X , d)\) is a complete separable metric space and \(T\) is an integral current on \((X , d)\). In 2011, Sormani and Wenger defined the intrinsic flat distance between two integral current spaces, and proved a compactness theorem with respect to that distance. We generalize these results to the setting of locally integral current spaces, which may have infinite mass, by introducing the pointed intrinsic flat distance.

SUMMARY: Translating solitons (translators for short) are hypersurfaces in Euclidean space defined as critical points of some weighted volume functional. The notion of stability for translators is naturally introduced with respect to the weighted measure.

In this talk, we show a topological result for stable translators. Roughly speaking, a stable translator must have simple shape. To be more precise, a complete stable translator admits no codimension one cycle which does not disconnect the translator. In particular, for a two dimensional surface case, this result means that a stable translator has no genus.

SUMMARY: We establish some Cheeger–Gromov–Taylor type theorems for forward complete Finsler manifolds via Bakry–Émery Ricci curvature. Our results generalize the Myers type theorem for forward complete Finsler manifolds due to S.-i. Ohta.

SUMMARY: In this talk, we give a necessary and sufficient condition of a LCK structure on a solvmanifold is a Vaisman structure. Thus, we see that Inoue surfaces and O–T manifolds has no Vaisman structures.

SUMMARY: Whether a constant scalar curvature Kähler (cscK) metric exists on a compact Kähler manifold is a question that attracted much attention in recent years. A continuity method for finding such metrics was proposed by X. X. Chen, which involves a twisted cscK metric that is an interesting object in its own right. We prove the openness for this continuity method, and quickly survey some relevant results.

SUMMARY: We study the Rarita–Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita–Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita–Schwinger fields. In the case of Calabi–Yau, hyperkähler, \(G_2\) and \(Spin(7)\) manifolds we find an identification of the kernel of the Rarita–Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita–Schwinger fields.