Submanifold Geometry and Lie Group Actions 2026

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.

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).

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.

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.

Date: 7-8, December, 2026