Waseda Young Researchers Meeting on Geometry 2025 in Fall

Abstract: A space-like submanifold of codimension 2 in a Lorentzian manifold is said to be marginally trapped if its mean curvature vector field is light-like.

When the ambient space is the Minkowski spacetime, a hypersurface with vanishing scalar curvature in the light-cone is a typical example of marginally trapped submanifolds. 

In this talk, I explain that a marginally trapped submanifold has a locally volume-maximizing property under specific conditions. 

Moreover, I will talk about my recent research.

Abstract: Li-Xia introduced a family of affine connections that connects the substatic condition in general relativity with 1-weighted Ricci curvature. On minimal hypersurfaces in manifolds with positive Ricci curvature, Choi-Wang established a lower bound for the first eigenvalue of the Laplacian. In this talk, we extend this estimate to the Laplacian associated with Li-Xia type affine connections.

Abstract: Given an edge-length parameter on a finite graph, we construct a vertex-weight and an edge-weight from it and define the corresponding graph Laplacian.

We consider a maximization of the first nonzero eigenvalue of the graph Laplacian over all edge-length parameters subject to a normalization.

We show that the supremum of the first nonzero eigenvalues infinite whenever the graph contains a cycle.

Abstract: Statistical structures can be regarded as a certain generalization of Riemannian structures. Statistical analogues of Kähler manifolds and almost Kähler manifolds have already been defined, and hypersurfaces of these manifolds have also been investigated. 

In this talk, I will present my recent results on hypersurfaces of nearly Kähler statistical manifolds, which may be seen as the statistical analogues of nearly Kähler manifolds.