Submanifold Geometry and Lie Group Actions 2026

Riku Kishida: "Zero mean curvature surfaces with singularities in 3-dimensional spaces with degenerate metrics"
The isotropic 3-space I^3 is the 3-dimensional real vector space R^3 with the degenerate metric of signature (0++). For a space-like surface in I^3, the mean curvature function can be defined in a natural way. In particular, zero mean curvature surfaces in I^3 can be regarded, via Weierstrass-type representation formulas, as intermediate objects between minimal surfaces in the Euclidean 3-space and maximal surfaces in the Lorentz-Minkowski 3-space. In this talk, we introduce a class of zero mean curvature surfaces with singularities in I^3, and we show that several Osserman-type inequalities can be obtained under certain assumptions on completeness and finite total curvature. We also present some examples which attain equality in these inequalities. If time permits, we also discuss the relationship with zero mean curvature surfaces in the 3-dimensional light-cone.

Keita Kunikawa: "Counterexamples to preservation of flat normal bundles under mean curvature flow"
The study of submanifolds in higher codimension differs substantially from the hypersurface case because of the complexity of the normal bundle. Two natural simplifying assumptions are the flatness of the normal bundle and the parallel principal normal condition. However, these conditions are not well suited to the study of mean curvature flow, since they are not preserved in general. We construct explicit counterexamples showing that the flatness of the normal bundle is not preserved under mean curvature flow. We also give an example showing that the parallel principal normal condition is not preserved, even though the normal bundle remains flat along the flow.

Takashi Sakai: "Polars and antipodal sets for the outer $3$-symmetric space $\mathbb{S}^7 \times \mathbb{S}^7$"
The Riemannian product of two symmetric spaces is again a symmetric space and naturally admits a $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric structure. Furthermore, as an exceptional case, $\mathbb{S}^7 \times \mathbb{S}^7$ admits a $3$-symmetric structure and an $S_3$-symmetric structure, where $S_3$ denotes the symmetric group of degree $3$. The Lie group $\mathrm{Spin}(8)$ acts transitively on $\mathbb{S}^7 \times \mathbb{S}^7$, and these symmetric structures arise from the triality automorphisms, which are outer automorphisms of $\mathrm{Spin}(8)$. In this talk, we discuss polars and antipodal sets for these symmetric structures on $\mathbb{S}^7 \times \mathbb{S}^7$. In particular, we show that a maximal antipodal set with respect to the $3$-symmetric structure consists of $3$ points and is preserved by the $S_3$-symmetric structure. This talk is based on joint work with Jost-Hinrich Eschenburg.

Yohei Sakurai: "Stability of weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound"
The aim of this talk is to present the validity of weighted Ricci curvature whose dimensional parameter is equal to ``1" in view of extrinsic geometric analysis. It has been observed that the 1-weighted Ricci curvature exhibits singular behavior from the view point of the Cheeger-Gromoll splitting theorem and affine geometry. Recently, it has been also pointed out that its non-negativity is equivalent to the so-called substatic condition in the context of the Lorentzian geometry via conformal change of metric. After I review such developments, I will introduce several geometric consequences concerning stable weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound. This talk is based on the joint work with Yasuaki Fujitani (University of Tokyo).

Haruka Sugai: "Classification of SNC-algebras in dimension five"
Every homogeneous manifold of negative curvature is known to be isometric to a Lie group with a left invariant metric. We define an SNC-algebra to be a Lie algebra which admits an inner product of strictly negative curvature. The classification of 4-dimensional SNC-algebras has been carried out previously. Recently, we obtained the classification results for 5-dimensional SNC-algebras. Moreover, using the conditions for an SNC-algebra to form a symmetric space as given by Heintze, we determined which among the 4-dimensional and 5-dimensional SNC-algebras correspond to symmetric spaces. Furthermore, for the 4-dimensional SNC-algebras that form symmetric spaces, we calculated the Riemann curvature tensors and discovered pairs of 4-dimensional SNC-algebras that are not isomorphic as Lie algebras but are isometric as Riemannian manifolds.

Tatsuya Tate: "Spectral geometry of the horizontal Laplacian on a locally Riemannian product submersion"
A locally Riemannian product submersion means a Riemannian submersion having totally geodesic fibers and integrable horizontal distribution, and hence it is very close to a Riemannian product space. However, as an example on the irrationally "tilted" torus indicates, the spectral properties of its horizontal Laplacian can be very complicated. In this talk, a theorem on the interpretation of horizontal Laplacian on a locally Riemannian product submersion into an operator on an infinite-rank flat vector bundle over a base manifold of the submersion will be explained. This theorem leads us to find interesting properties of the spectrum of the horizontal Laplacian, some of which will be also explained.

Kazuyuki Hasegawa: "Quaternionic complex manifolds and fixed-point sets of $S^1$-actions"
The study fixed-point sets of $S^1$-actions and compatible complex structures on quaternionic manifolds. If the first Chern class of a fixed-point component is nontrivial, then there is no hypercomplex structure on the quaternionic manifold containing the prescribed compatible complex structure induced from a twistor function on that component. Furthermore, we determine the connected components of fixed-point sets arising from quaternionic $S^{1}$-actions on the quaternionic projective space. Consequently, we generalize Pontecorvo's example $\mathrm{SO}^{\ast}(2n+2)/\mathrm{SO}^{\ast}(2n) \times \mathrm{SO}^{\ast}(2)$. This talk is based on ``Quaternionic complex manifolds and fixed-point sets of $S^{1}$-actions, preprint, arXiv:2603.02755''.

Naotoshi Fujihara: "Isotropy-invariant self-shrinking hypersurfaces and geodesics"
It is known that self-shrinking hypersurfaces in Euclidean space are minimal hypersurfaces with respect to a suitable conformally flat metric. Moreover, rotationally symmetric self-shrinking hypersurfaces correspond to geodesics on a surface with a suitable conformally flat metric. We then regard the tangent space of a symmetric space as a Euclidean space and consider the linear isotropy action on it. In this talk, we discuss the correspondence between self-shrinking hypersurfaces invariant under this isotropy action and geodesics on a surface with a suitable conformally flat metric. This talk is based on joint work with Naoyuki Koike (Tokyo University of Science).

Fumika Mizoguchi: "Geometric structures on nilpotent and solvable Lie algebras obtained by quivers"
In differential geometry, an important problem is to study when a given Lie group admits special left-invariant geometric structures. In particular, solvable Lie groups are interesting objects. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles. We also showed that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this talk, we construct solvable Lie algebras by adding vertices as paths of length zero. We discuss the existence of special left-invariant geometric structures, including Ricci solitons, on the corresponding simply-connected Lie groups. Furthermore, we investigate the relationship between the existence of such geometric structures and the combinatorial properties of the underlying quivers.

Katsuhiro Moriya: "Representation formulas for minimal timelike surfaces in four-dimensional pseudo-Euclidean space"
We study minimal timelike surfaces in the four-dimensional pseudo-Euclidean space of index two. By constructing representation formulas for local null curves, we obtain explicit parametrizations of local minimal timelike surfaces without integration. As a special case, the method also gives a representation formula for null curves in three-dimensional Minkowski space.

Submanifold Geometry and Lie Group Actions 2026

Date: 7-8, December, 2026

Venue: Tokyo University of Science, Kagurazaka Campus

7, December: Morito Memorial Hall, Forum No. 1

8, December: Morito Memorial Hall, Forum No. 2

Invited Speakers (confirmed)

Program:

7 December

8 December

Title and Abstract