Shin-ichi Ohta’s research while affiliated with RIKEN Center for Advanced Intelligence Project and other places

In proper, geodesic Gromov hyperbolic spaces, we investigate discrete-time gradient flows via the proximal point algorithm for unbounded Lipschitz convex functions. Assuming that the target convex function has negative asymptotic slope along some ray (thus unbounded below), we first prove the uniqueness of such a negative direction in the boundary at infinity. Then, using a contraction estimate for the proximal (resolvent) operator established in our previous work, we show that the discrete-time gradient flow from an arbitrary initial point diverges to that unique direction of negative asymptotic slope. This is inspired by and generalizes results of Karlsson-Margulis and Hirai-Sakabe on nonpositively curved spaces and a result of Karlsson concerning semi-contractions on Gromov hyperbolic spaces.

A timelike splitting theorem for Finsler spacetimes was previously established by the third author, in collaboration with Lu and Minguzzi, under relatively strong hypotheses, including the Berwald condition. This contrasts with the more general results known for positive definite Finsler manifolds. In this article, we employ a recently developed strategy for proving timelike splitting theorems using the elliptic p-d'Alembertian. This approach, pioneered by Braun, Gigli, McCann, S\"amann, and the second author, allows us to remove the restrictive assumptions of the earlier splitting theorem. For timelike geodesically complete Finsler spacetimes, we establish a diffeomorphic splitting. In the specific case of Berwald spacetimes, we show that the Busemann function generates a group of isometries via translations. Furthermore, for Berwald spacetimes, we extend these splitting theorems by replacing the assumption of timelike geodesic completeness with global hyperbolicity. Our results encompass and generalize the timelike splitting theorems for weighted Lorentzian manifolds previously obtained by Case and Woolgar-Wylie.

We generalize Gr\"unbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $\Omega$ and the Busemann function associated with any straight line, the volume of the intersection of $\Omega$ and any sublevel set of the Busemann function that contains a barycenter of $\Omega$ is bounded from below in terms of N. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature R i c N \mathrm {Ric}_N is bounded from below by a real number K K in every timelike direction satisfies the timelike curvature-dimension condition T C D q ( K , N ) \mathrm {TCD}_q(K,N) for all q ∈ ( 0 , 1 ) q\in (0,1) . The converse and a nonpositive-dimensional version ( N ≤ 0 N \le 0 ) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q q -Lorentz–Wasserstein distance as well as the characterization of q q -geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case.

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from below by a real number K in every timelike direction satisfies the timelike curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also shown. Our discussion is based on the solvability of the Monge problem with respect to the q-Lorentz-Wasserstein distance as well as the characterization of q-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case.

We show an analog of the Lorentzian splitting theorem for weighted Lorentz–Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the η-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar–Wylie into a single framework.

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.