研究情報

It is an open problem posed by M. Gromov to characterize those metric spaces that admit a distance-preserving embedding into a $\mathrm{CAT}(0)$ space. As a partial result of this problem, it is known that a $5$-point metric space admits a distance-preserving embedding into a $\mathrm{CAT}(0)$ space if and only if any 4 points in it satisfy the family of inequalities called the weighted quadruple inequalities. On the other hand, it is also known that the validity of the weighted quadruple inequalities does not suffice for a 6-point metric space to admit a distance-preserving embedding into a $\mathrm{CAT}(0)$ space. This means that there exist inequalities that hold true for any 6 points in any $\mathrm{CAT}(0)$ space but do not follow from the weighted quadruple inequalities. In this talk, we establish the first examples of such inequalities.

Given a Riemannian submersion $(M, g) \to (B, j)$ each of whose fiber is connected and totally geodesic, we consider a certain $1$-parameter family of Riemannian metrics $(g_{t})_{t} > 0$ on $M$, which is called the canonical variation. Let $\lambda_{1}(g_{t})$ be the first positive eigenvalue of the Laplace–Beltrami operator $\Delta_{g_{t}}$ and $\operatorname{Vol} (M, g_{t})$ the volume of $(M, g_{t})$. The main theorem of this talk is that if each fiber is Einstein and $(M, g)$ satisfies a certain condition about its Ricci curvature, then the scale-invariant quantity $\lambda_{1}(g_{t}) \operatorname{Vol} (M, g_{t})^{2 / \dim M}$ goes to $\infty$ with $t$. In the talk, we will see as many examples to which this theorem can be applied as time permits. This talk is based on my preprint arXiv:2411.17078.