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Since the year 2000, we have witnessed several outstanding results in geometry that have solved long-standing problems such as the Poincaré conjecture, the Yau–Tian–Donaldson conjecture, and the Willmore conjecture. There are still many important and challenging unsolved problems including, among others, the Strominger–Yau–Zaslow conjecture on mirror symmetry, the relative Yau–Tian–Donaldson conjecture in Kähler geometry, the Hopf conjecture, and the Yau conjecture on the first eigenvalue of an embedded minimal hypersurface of the sphere. For the younger generation to approach such problems and obtain the required techniques, it is of the utmost importance to provide them with up-to-date information from leading specialists.The geometry conference for the friendship of China and Japan has achieved this purpose during the past 10 years. Their talks deal with problems at the highest level, often accompanied with solutions and ideas, which extend across various fields in Riemannian geometry, symplectic and contact geometry, and complex geometry.



Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group

In this paper, we give a representation formula for Legendrian surfaces in the 5-dimensional Heisenberg group \(\mathfrak {H}^5\), in terms of spinors. For minimal Legendrian surfaces especially, such data are holomorphic. We can regard this formula as an analogue (in Contact Riemannian Geometry) of Weierstrass representation for minimal surfaces in \(\mathbb {R}^3\). Hence for minimal ones in \(\mathfrak {H}^5\), there are many similar results to those for minimal surfaces in \(\mathbb {R}^3\). In particular, we prove a Halfspace Theorem for properly immersed minimal Legendrian surfaces in \(\mathfrak {H}^5\).
Reiko Aiyama, Kazuo Akutagawa

Gluing Principle for Orbifold Stratified Spaces

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth structure on the stratified space. As an application, we provide an orbifold structure on the coarse moduli space \(\overline{M}_{g, n}\) of stable genus g curves with n-marked points. Using the gluing theory for \(\overline{M}_{g, n} \) associated to horocycle structures, there is a natural orbifold gluing structure on \(\overline{M}_{g, n}\). We show this gluing atlas can be refined to provide a good orbifold gluing atlas and hence a smooth orbifold structure on \(\overline{M}_{g,n}\). This general gluing principle will be very useful in the study of the gluing theory for the compactified moduli spaces of stable pseudo-holomorphic curves in a symplectic manifold.
Bohui Chen, An-Min Li, Bai-Ling Wang

Applications of the Affine Structures on the Teichmüller Spaces

We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of Calabi–Yau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact Calabi–Yau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about certain period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of Calabi–Yau type manifolds to their period domains is proved and applied to the cases of K3 surfaces, cubic fourfolds, and hyperkähler manifolds.
Kefeng Liu, Yang Shen, Xiaojing Chen

Critical Points of the Weighted Area Functional

In this survey, we discuss critical points of functionals by various aspects. We review properties of critical points of weighted area functional, that is, self-shrinkers of mean curvature flow in Euclidean spaces and examples of compact self-shrinkers are discussed. We also review properties of critical points for weighted area functional for weighted volume-preserving variations, that is, \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean spaces.
Qing-Ming Cheng

A New Look at Equivariant Minimal Lagrangian Surfaces in

In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.
Josef F. Dorfmeister, Hui Ma

A Survey on Balanced Metrics

This survey will focus on the existences of balanced metrics on a compact hermitian manifolds.
Jixiang Fu

Can One Hear the Shape of a Group?

The iso-spectrum problem for marked lengnth spectrum for Riemannian manifolds of negative curvature has a rich history. We rephrased the problems for metrics on discrete groups, discussed its connection to a conjecture by Margulis, and proved some results for “total relatively hyperbolic groups” in Koji Fujiwara, Journal of Topology and Analysis, 7(2), 345–359 (2015). This is a note from my talk on that paper and mainly discuss the connection between Riemannian geometry and group theory, and also some questions.
Koji Fujiwara

Differential Topology Interacts with Isoparametric Foliations

In this note, we discuss the interactions between differential topology and isoparametric foliations, surveying some recent progress and open problems.
Chao Qian, Jianquan Ge

Unobstructed Deformations of Generalized Complex Structures Induced by Logarithmic Symplectic Structures and Logarithmic Poisson Structures

We shall introduce the notion of \(C^{\infty }\) logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a \(C^{\infty }\) logarithmic symplectic structure has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of type changing loci if the type changing loci are smooth. Complex surfaces with smooth effective anti-canonical divisors admit unobstructed deformations of generalized complex structures such as del pezzo surfaces and Hirzebruch surfaces. We also give some calculations of Poisson cohomology groups on these surfaces. Generalized complex structures \(\mathscr {J}_m\) on the connected sum \((2k-1)\mathbb {C}P^2\# (10k-1)\overline{{\mathbb {C}P^2}}\) are induced by \(C^{\infty }\) logarithmic symplectic structures modulo the action of b-fields and it turns out that generalized complex structures \(\mathscr {J}_m\) have unobstructed deformations of dimension \(12k+2m-3\).
Ryushi Goto

The Symplectic Critical Surfaces in a Kähler Surface

In this paper, we study the functional \(L_\beta =\int _\varSigma \frac{1}{\cos ^\beta \alpha }d\mu ,~\beta \ne -1\) in the class of symplectic surfaces. We derive the Euler-Lagrange equation. We call such a critical surface a \(\beta \)-symplectic critical surface. When \(\beta =0\), it is the equation of minimal surfaces. When \(\beta \ne 0\), a minimal surface with constant Kähler angle satisfies this equation, especially, a holomorphic curve or a special Lagrangian surface satisfies this equation. We study the properties of the \(\beta \)-symplectic critical surfaces.
Xiaoli Han, Jiayu Li, Jun Sun

Some Evolution Problems in the Vacuum Einstein Equations

We discuss two problems in the evolution of the vacuum Einstein equations. The first one is about the formation of trapped surface, and the second one is about the characteristic problems with initial data on complete null cones.
Junbin Li, Xi-Ping Zhu

Willmore 2-Spheres in : A Survey

We give an overview of the classification problem of Willmore 2-spheres in \(S^n\), and report the recent progress on this problem when \(n=5\) (or even higher). We explain two main ingredients in our work. The first is the adjoint transform of Willmore surfaces introduced by the first author, which generalizes the dual Willmore surface construction. The second is the DPW method applied to Willmore surfaces whose conformal Gauss map is well-known to be a harmonic map into a non-compact symmetric space (a joint work of Dorfmeister and the second author). We also sketch a possible way to classify all Willmore 2-spheres in \(S^n\).
Xiang Ma, Peng Wang

The Yau-Tian-Donaldson Conjecture for General Polarizations, I

In this paper, some \(C^0\) boundedness property (BP) is introduced for balanced metrics on a polarized algebraic manifold (XL). Then by assuming that (XL) is strongly K-stable in the sense of [8], we shall show that the balanced metrics have (BP). In a subsequent paper [10], this property (BP) plays a very important role in the study of the Yau-Tian-Donaldson conjecture for general polarizations.
Toshiki Mabuchi

Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

We define cuspidal curvature \(\kappa _c\) (resp. normalized cuspidal curvature \(\mu _c\)) along cuspidal edges (resp. at a swallowtail singularity) in Riemannian 3-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product \(\kappa _\varPi ^{}\) called the product curvature (resp. \(\mu _\varPi ^{}\) called normalized product curvature) of \(\kappa _c\) (resp. \(\mu _c\)) and the limiting normal curvature \(\kappa _\nu \) is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of \(\kappa _\varPi ^{}\) when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.
L. F. Martins, K. Saji, M. Umehara, K. Yamada

The Calabi Invariant and Central Extensions of Diffeomorphism Groups

Let D be a closed unit disc in dimension two and G the group of symplectomorphisms on D. Denote by \(G_{\partial }\) the group of diffeomorphisms on the boundary \(\partial D\) and by \(G_{\mathrm {rel}}\) the group of relative symplectomorphisms. There exists a short exact sequence involving with those groups, whose kernel is \(G_{\mathrm {rel}}\). On such a group \(G_{\mathrm {rel}}\) one has a celebrated homomorphism called the Calabi invariant. By dividing the exact sequence by the kernel of the Calabi invariant, one obtains a central \(\mathbb R\)-extension, called the Calabi extension. We determine the resulting class of the Calabi extension in \(H^2( G_{\partial };\mathbb R)\) and exhibit a transgression formula that clarify the relation among the Euler cocycle for \(G_{\partial }\), the Thom class and the Calabi invariant.
Hitoshi Moriyoshi

Concentration, Convergence, and Dissipation of Spaces

We survey some parts of Gromov’s theory of metric measure spaces [6, Sect. 3.\(\frac{1}{2}\)], and report our recent works [1417], focusing on the asymptotic behavior of a sequence of spaces with unbounded dimension.
Takashi Shioya

The Space of Left-Invariant Riemannian Metrics

Geometry of left-invariant Riemannian metrics on Lie groups has been studied very actively. We have proposed a new framework for studying this topic from the viewpoint of the space of left-invariant metrics. In this expository paper, we introduce our framework, and mention two results. One is a generalization of Milnor frames, and another is a characterization of solvsolitons of dimension three in terms of submanifold geometry.
Hiroshi Tamaru

Futaki Invariant and CM Polarization

This is an expository paper. We will discuss various formulations of Futaki invariant and its relation to the CM line bundle. We will discuss their connections to the K-energy. We will also include proof for certain known results which may not have been well presented or less accessible in the literature. We always assume that M is a compact Kähler manifold. By a polarization, we mean a positive line bundle L over M, then we call (ML) a polarized manifold.
Gang Tian
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