Differential Geometry Seminar (2025)

A new framework of zero mean curvature surfaces in the isotropic 3-space

The isotropic 3-space I³ is the 3-dimensional real vector space with the degenerate inner product of signature (0++). For a space-like surface in I³, the mean curvature function can be defined in a natural way. A space-like surface whose mean curvature function vanishes everywhere is called a zero mean curvature surface. A zero mean curvature surface in I³ can be regarded, via Weierstrass-type representation formulas, as the intermediate between a minimal surface in the Euclidean 3-space and a maximal surface in the Lorentz-Minkowski 3-space. As in the case of maximal surfaces, there exist many examples of zero mean curvature surfaces in I³ with singularities.
In this talk, we introduce ZMC-faces as a class of zero mean curvature surfaces in I³ which admit certain kinds of singularities. As an application of ZMC-faces, we show that several Osserman-type inequalities can be obtained under certain assumptions on both completeness and finiteness of the total curvature. We also present some examples which attain the equality in these inequalities.

Totally geodesic submanifolds and symmetry

Totally geodesic submanifolds play a fundamental role in Riemannian geometry and they are closely related to symmetry. For this reason, their study is particularly rich in highly symmetric settings, such as homogeneous and symmetric spaces, providing deep insights into their underlying geometric structures and properties. In this talk, I will present a broad overview of recent developments in the classification, construction, and applications of totally geodesic submanifolds in these spaces. Emphasizing both classical and new results, I will discuss their connections to Lie group theory and rigidity phenomena, as well as highlight open questions and potential directions for future research.

Analytic torsion for irreducible holomorphic symplectic fourfolds with antisymplectic involution

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Compact Kähler Manifolds with Nef Anticanonical Bundle

In this talk, we present recent results obtained in collaboration with Shin-ichi Matsumura, Juanyong Wang, and Qimin Zhang, concerning compact Kähler manifolds whose anticanonical bundle is nef. It has been conjectured that every compact Kähler manifold with nef anticanonical bundle admits a locally trivial fibration over a Calabi–Yau manifold. Significant progress has been made in the projective setting, in particular through the works of Cao and Höring, who established the structure of projective varieties with nef anticanonical bundles. However, extending these results to the compact Kähler setting poses substantial challenges. The recent breakthrough of Wenhao Ou, who resolved a conjecture proposed by Boucksom, Demailly, Păun, and Peternell, provides new tools to address these difficulties. Building on these developments, we aim to show that the conjecture holds in full generality for compact Kähler manifolds.

The Riesz energy function (Brylinski beta function) and residues of manifolds

　Let X be a compact submanifold of a Euclidean space, either closed or of the same dimension as the ambient space. The Riesz energy function or the Brylinski beta function of X, B_X(z), is a function of a complex variable z defined as the regularization of the integral on the product space X times X of the distance between pairs of points on X to the power z. It is a meromorphic function only with simple poles. The residues of this function is called the residues of the manifold X. I will introduce properties of the function and residues.
　B_X(z) is equivalent to the interpoint distance distribution, and the chord length distribution when X is convex, which have been studied in integral geometry via Mellin transform. When z=-2 dim X, B_X(z) or the residue is invariant under Moebius transformations. I will also consider a problem to what extent a space can be identified by B_X(z).

Gromov-Hausdorff convergence of Delzant polytopes and toric symplectic manifolds

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Delzant type theorem for subtorus orbit closures in symplectic toric manifolds

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Extremal Kähler metrics and destabilizers for relative K-polystability of toric varieties

It was conjectured by Székelyhidi that a polarized manifold admits an extremal Kähler metric in the class of polarization if and only if it is relatively K-polystable. Furthermore, the folklore conjecture states that every toric Fano manifold admits an extremal Kähler metric in its first Chern class. For a given toric Fano manifold X, we provide a destabilizing convex function on the corresponding moment polytope P to clarify the relative K-unstability of X. Applying this criteria into a certain toric Fano manifold, we prove that there exists a toric Fano manifold of dimension 10 that does not admit an extremal Kähler metric. This talk is based on joint work with B. Zhou, and another recent work with D. Hwang and H. Sato.

On s-commutative sets in real Grassmannian manifolds and representations of Clifford algebras

An s-commutative set in a quandle or a symmetric space is a set in which, for any two points, the point symmetries at those points are commutative. This notion is a generalization of the antipodal set, which was introduced by Hiroshi Tamaru et al. Moreover we expect that they have geometric information about the quandles or symmetric spaces. Although, for the s-commutative sets, several concrete examples are known, the details of them are not clear. In this talk, I will present a construction of s-commutative sets in real Grassmannian manifolds derived from representations of Clifford algebras. In addition, I will also discuss the relationship between our results and the classification of maximal antipodal sets in real Grassmannian manifolds given by Makiko Tanaka and Hiroyuki Tasaki.

A characterization of Oeljeklaus-Toma manifolds in LCK geometry

Oeljeklaus–Toma (OT) manifolds are known as examples of complex manifolds that do not admit a Kähler metric, and they are regarded as higher-dimensional analogues of Inoue surfaces. OT manifolds are solvmanifolds constructed from number-theoretic data, and some of them admit locally conformally Kähler (LCK) metrics. In this way, a large number of examples of solvmanifolds equipped with LCK metrics have been obtained, and OT manifolds have been actively studied as important examples in LCK geometry. Although their construction may seem intricate, aside from some simple examples, OT manifolds are the only known solvmanifolds admitting LCK metrics. In this talk, I will show that if a certain class of solvmanifolds admits an LCK metric, then it is essentially an OT manifold. Since number theory naturally arises from geometric constraints in our setting, this result suggests that number-theoretic arguments are indispensable in the construction of certain classes of solvmanifolds. This talk is based on the preprint arXiv:2502.12500.