arXiv subjects
Publications and source records attributed to Kenshiro Tashiro.
We study the structure of Busemann spaces with measures satisfying the measure contraction property (MCP). The main results are rigidity theorems and structure theorems under the assumption of geodesic completeness or non-collapse. The appendix contains some observations on the tangent cones of geodesically complete Busemann spaces.
In this paper, we find a gap between the lower bound of the Bakry-\'Emery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-\'Emery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<\beta<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_\beta)$ with $w_\beta=\prod_{i<j}^n |x_i-x_j|^\beta$ and any $N\in[n+\frac{\beta}{2}n(n-1),+\infty]$, $\beta \ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold; $0 < \beta < 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-\'Emery curvature bound in the small inverse temperature regime.
We provide new examples of sub-Riemannian manifolds with boundary equipped with a smooth measure that satisfy the $\mathsf{RCD}(K , N)$ condition. They are constructed by equipping the half-plane, the hemisphere and the hyperbolic half-plane with a two-dimensional almost-Riemannian structure and a measure that vanishes on their boundary. The construction of these spaces is inspired from the geometry of the $\alpha$-Grushin plane.
The curvature exponent $N_{\mathrm{curv}}$ of a metric measure space is the smallest number $N$ for which the measure contraction property $\mathsf{MCP}(0,N)$ holds. In this paper, we study the curvature exponent of sub-Finsler Heisenberg groups equipped with the Lebesgue measure. We prove that $N_{\mathrm{curv}} \geq 5$, and the equality holds if and only if the corresponding sub-Finsler Heisenberg group is actually sub-Riemannian. Furthermore, we show that for every $N\geq 5$, there is a sub-Finsler structure on the Heisenberg group such that $N_{\mathrm{curv}}=N$.
We study the horofunction boundary of finitely generated nilpotent groups, and the natural group action on it. More specifically, we prove the followings results: For discrete Heisenberg groups, we classify the orbits of Busemann points. As a byproduct, we observe that the set of orbits is finite and the set of Busemann points is countable. Furthermore, using the approximation with Lie groups, we observe that the entire horoboundary is uncountable. For the discrete Cartan group, we exhibit an continuum of Busemann points, disproving a conjecture of Tointon and Yadin. As a byproduct, we prove that the group acts non-trivially on its reduced horoboundary, disproving a conjecture of Bader and Finkelshtein.
In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure.
In this paper, we study a non-collapsed Gromov--Hausdorff limit of a sequence of compact Heisenberg manifolds with sub-Riemannian metrics. In the case of strictly sub-Riemannian case, we show that if a sequence has an upper bound of the diameter and a lower bound of Popp's measure, then it has a convergent subsequence in the Gromov--Hausdorff topology, and the limit is isometric to a compact Heisenberg manifold of the same dimension. The same conclusion is also shown for Riemannian case with the additional assumption on Ricci curvature lower bounds.
We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with an $\ell^p$-sub-Finsler norm. We show that for $p\in(2,\infty]$, the $\ell^p$-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if $p\in(1,2)$, then it satisfies the measure contraction property $\mathsf{MCP}(K, N)$ if and only if $K \leq 0$ and $N \geq N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$ being the H\"older conjugate of $p$). We also prove that the geodesic dimension of the $\ell^p$-Heisenberg group is $\min(2q+2,5)$ for $p\in[1,\infty)$. As a consequence, we provide the first example of a metric measure space where there is a gap between the curvature exponent and the geodesic dimension.
In this paper, we give a systolic inequality for a quotient space of a Carnot group $Γ\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $Γ\backslash G$ is less than ${\rm Cvol}(Γ\backslash G)^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=\bigoplus V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.
In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the Popp's volume or the minimal Popp's volume converges to zero. In the appendix, we give the systolic inequality on sub-Riemannian Heisenberg manifolds, and observe that the exponent of the total measure is equal to the inverse of the Hausdorff dimension.
We study the speed of convergence to the asymptotic cone for Cayley graphs of nilpotent groups. Burago showed that $\{(\mathbb{Z}^d, \frac{1}{n} ρ,id)\}_{n\in\mathbb{N}}$ converges to $(\mathbb{R}^d,d_{\infty},id)$ and its speed is $O(\frac{1}{n})$ in the sense of Gromov-Hausdorff distance. Later Breuillard and Le Donne gave estimates for non-abelian cases, and constructed an example whose speed of convergence is slower than $O(\frac{1}{\sqrt{n}})$. For $2$-step nilpotent groups, we show that if the Mal'cev completion is non-singular, then the speed of convergence is $O(\frac{1}{n})$ for any choice of generating set. In terms of subFinsler geometry, this condition is also equivalent to the absence of abnormal geodesics on the asymptotic cone.