2017年度秋季総合分科会(於:山形大学)

SUMMARY: We will talk about the relationship between the existence of “canonical” Kähler metrics and algebraic geometric stabilities of polarized manifolds. It is conjectured that the existence of constant scalar curvature Kähler metrics will be equivalent to the notion of K-polystability, which is known as the Yau–Tian–Donaldson conjecture. For the case of Kähler–Einstein metrics on Fano manifolds, it was solved affirmatively by Chen–Donaldson–Sun. However, for general polarizations, the above conjecture is still open. In this talk, we shall discuss on the recent developement for this problem and some versions. (for extremal Kähler metrics, generalized Kähler–Einstein metrics, etc.)

SUMMARY: For many years I have been interested in conformal differential geometry and projective differential geometry. In this talk I would like to explain what these geometries mean at the present day and what can be expected in the future. I am now, as of June 2017, planning the talk and it will be concerned chiefly with a new conformal invariant which is similar to Yamabe’s conformal invariant in many respects. In addition a conjecture, a sphere theorem, will be presented. If time permits I touch upon projective differential geometry and discuss some complements to Weyl’s setting. This talk as a whole is a derivation from H. Weyl’s “Reine Infinitesimalgeometrie” and subsequent developments by K. Yano, H. Yamabe and M. Obata.

SUMMARY: For a metric space \((X,d)\), the Gromov–Hausdorff limit of \((X,a_n d)\) as \(a_n\to 0\) is called the tangent cone at infinity of \((X,d)\). Although the tangent cone at infinity always exists if \((X,d)\) is a complete Riemannian manifold with nonnegative Ricci curvature, the uniqueness does not hold in general. Colding and Minicozzi showed the uniqueness under the assumption that \((X,d)\) is a Ricci-flat manifold satisfying some additional conditions. In this talk, I will explain some examples of noncompact complete hyper-Kähler manifolds who have several tangent cones at infinity, and determine the moduli spaces of them.

SUMMARY: We construct several invariants of tuples of commutative diffeomorphisms on a 4-manifold using the Seiberg–Witten theory for families. They are generalizations of Ruberman’s invariants of diffeomorphisms on a 4-manifold using 1-parameter gauge theory. We also give an application to the spaces of positive scalar curvature metrics for 4-manifolds which cannot be studied using Ruberman’s original invariant.

SUMMARY: Inspired by the Lichnerowicz–Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants \(\{\Omega _k(g)\}\) for closed Riemannian manifolds. The value of \(\Omega _k(g)\) sharply reflects the spherical part of the manifold. Indeed, \(\Omega _1(g)\) and \(\Omega _2(g)\) characterize the standard sphere.

SUMMARY: Let \(G\) be a connected compact Lie group. A \(G\)-invariant differential operator on a compact \(G\)-manifold is said to be transversally elliptic if it is elliptic in the directions transversal to the \(G\)-orbits. In this talk we study the heat operator of a transversally elliptic operator. After we review the spectral properties of a transversally elliptic operator, we define and investigate the character, that is a distribution on \(G\) generalizing the trace of the heat operator to the \(G\)-equivariant case.

SUMMARY: In this talk, we extend Roe’s cyclic \(1\)-cocycle to relative settings. We also state two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of [arXiv:1411.6090, Theorem 3.3].

SUMMARY: An open convex cone \(\Omega \subset \mathbb {R}^n\) is said to be regular if \(\Omega \) does not contain any full straight lines. We obtain vanishing theorems of \(L^2\)-cohomology groups \(L^2H^{p,q}_{\bar {\partial }}(\Omega ,g)\) on a regular convex cone \(\Omega \subset \mathbb {R}^n\) with the Cheng–Yau metric \(g\) for \(p+q>n\) or \(p>q\).

SUMMARY: We give examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi–Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space due to Takagi.

SUMMARY: We show that a parallel differential form \(\Psi \) of even degree on a Riemannian manifold allows to define a natural differential both on \(\Omega ^*(M)\) and \(\Omega ^*(M, TM)\), defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel \(4\)-form on a \(G_2\)- and \({\rm Spin}(7)\)-manifold, respectively. We calculate the cohomology groups of \(\Omega ^*(M)\) and give a partial description of the cohomology of \(\Omega ^*(M, TM)\).

SUMMARY: We define two parabolic flows on almost complex manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow respectively on complex manifolds. We study the relationship between these parabolic evolution equations on a compact almost Hermitian manifold.

SUMMARY: Leung, Yau and Zaslow defined deformed Hermitian Yang–Mills connections and gave a formal way to convert special Lagrangian submanifolds in \(X\) to deformed Hermitian Yang–Mills connections on \(W\). In their paper, two conditions were assumed for simplicity. One is that \(X\) and \(W\) are actually lattice quotients of tangent and cotangent bundle of some common open subset \(B\) in \(\mathbb {R}^{m}\). Another is that each Lagrangian submanifold can be written as a graph of a section of \(X\to B\). In this talk, a way to glue their argument on a tropical manifold will be explained and a mild condition for Lagrangians will be mentioned.

SUMMARY: We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.

SUMMARY: The Miyaoka–Yau inequality is an inequality for Chern classes. In this talk, we prove it for compact Kähler manifold with semipositive canonical bundle.

SUMMARY: Anti-canonically balanced metrics are quantization of Kähler–Einstein metrics on Fano manifolds. We introduce a new algebro-geometric stability on Fano manifolds and show that the existence of anti-canonically balanced metrics implies our stability. The relation between our stability and others is also discussed.

SUMMARY: On a Fano manifold \(M\), the conical Kähler–Ricci flow (CKRF) evolves Kähler metrics while preserving cone singularities along an effective divisor \(D\). When \(D\) is smooth and \(M\) admits a conical Kähler–Einstein metric, the limiting behavior of CKRF is studied by Liu–Zhang. However, allowing \(D\) to have simple normal crossing support changes the whole situation. In this talk, we consider the case when \(D\) is simple normal crossing and \(M\) admits a conical Kähler–Ricci soliton. In order to study the limiting behavior of CKRF, we construct the regularized flow of CKRF modified by a holomorphic vector field from the view point of the gradient flow interpretation with respect to the modified log/twisted Mabuchi \(K\)-energy.

SUMMARY: Percolation process is a kind of probability theory. Let \(G=(V,E)\) be a connected graph, and fix a parameter \(p \in [0,1]\), then we consider each edge \(e \in E\) to be open with probability \(p\) independently. We have a subgraph of \(G\) which consisted only of open edges, it has some connected components. In this talk, we discuss the relationship between the number of connected components and Cheeger constant.

SUMMARY: We consider orthocenters of simplices of the unit sphere of the \(n\)-dimensional Euclidean space. For \(n=3\), orthocenters always exist for all simplices, but for \(n\geqq 4\), they do not necessarily exist. Moreover, unlike the case of the Euclidean space, it is possible that there exist infinite numbers of orthocenters. In this talk, we give characterizations of the existence and the uniqueness of orthocenters.

SUMMARY: In this talk, stimulated by Ohta and Takatsu, and Wylie, we shall establish some new compactness theorems for complete Riemannian manifolds via \(m\)-Bakry–Émery and \(m\)-modified Ricci curvatures with negative \(m\). Our compactness theorems may be considered as natural generalizations of the classical compact theorems due to Ambrose, Galloway, and Cheeger–Gromov–Taylor.

SUMMARY: We consider semi-Riemannian submersions \(\pi : (E, g) \rightarrow (B, -g_{B}) \) under the condition with \((B, g_{B})\) Riemannian, the fiber closed Riemannian, and the horizontal distribution integrable. Then we prove that, if the non-spacelike geodesically complete semi-Riemannian manifold \(E\) has some positivity of curvature, then the fundamental group of the fiber is finite. Moreover we construct an example of semi-Riemannian submersions with some positivity of curvature, non-integrable horizontal distribution, and the finiteness of the fundamental group of the fiber.

SUMMARY: Let \(g\) be a Riemannian metric on a manifold \(M\). We call a Finsler metric \(F(x, y)=\sqrt {g(x)(y , y)} + \omega (y)\) a Randers metric where \(\omega \) is a 1-form on \(M\). Two Finsler metrics are said to be pointwise projectively related if they have the same geodesics as point sets. If the 1-form \(\omega \) is closed, then \(F\) is pointwise projectively related to \(g\). We see that the invariance of the cut loci w.r.t \(g\) and \(F\) implies the exactness of a closed 1-form \(\omega \).

SUMMARY: In present presentation, we attempt to identify a hidden smooth surface for a given discrete surface by providing a convergence theorem of the sequence of subdivisions of a network. We prove the sequence of the Goldberg–Coxeter subdivisions of a trivalent network realized in 3-dim Euclidean space by harmonic maps are consisting of a Cauchy sequence in the Hausdorff topology. As an application, we study the Mackay Crystal and estimate the convergence of subdivisions and their normal vectors.

SUMMARY: We first give a simple criterion for (the lowest order) isotropy of a spherical minimal immersion in terms of orthogonality relations in the third (ordinary) derivative of the image curves. This is then applied in the main result of this talk which gives a full characterization of isotropic \(SU(2)\)-equivariant spherical minimal immersions of \(S^3\) into the unit sphere of real and complex \(SU(2)\)-modules. Specific examples include the polyhedral minimal immersions of which the icosahedral minimal immersion is isotropic whereas its tetrahedral and octahedral cousins are not.

SUMMARY: This talk gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type. We see that there are very fruitful ruled minimal surfaces in four dimensional Minkowski space or four dimensional pseudo-Euclidean space with neutral metric, i.e. having index 2. In particular, it should be remarkable that some of those ruled minimal surfaces are embedded in three dimensional subspace with degenerate metric of pseudo-Euclidean space.

SUMMARY: We determine three-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with nondiagonalizable Tchebychev operator.

SUMMARY: In the continuous case, a timelike immersion with vanishing mean curvature in 3-dimensional Minkowski space is called a timelike minimal surface. Timelike minimal surfaces are highly related to linear and nonlinear wave equations. In this talk we briefly introduce a theory of discrete timelike surfaces. In particular, by a reparametrizion of discrete surfaces, we show that each coordinate function of a discrete timelike minimal surface satisfies a discrete wave equation. This result provides not only the geometric meaning of special solutions for a discrete wave equation but also a new representation formula for discrete timelike minimal surfaces.

SUMMARY: The total mixed curvature of a curve in \(E^3\) is defined as the integral of \(\sqrt {\kappa ^2+\tau ^2}\), where \(\kappa \) is the curvature and \(\tau \) is the torsion. The total mixed curvature is the length of the spherical curve defined by the principal normal vector field. We study the infimum of the total mixed curvature in a set of curves, where the endpoints and the principal normal vectors at the endpoints are prescribed. In our previous works, similar problems have been studied for the unit tangent vector and for the binormal vector.

SUMMARY: Geometric counterparts of the Miura transformation between the Korteweg–de Vries (KdV) equation and the defocusing modified KdV equation are given, by using the fact that the KdV equation arises from certain time-evolutions of equicentroaffine curves and the defocusing modified KdV equation from those of curves on a Minkowski plane and on a two-dimensional de Sitter space.