Proper Geodesic Space Research Articles

Principal eigenvalue problem for infinity Laplacian in metric spaces

Abstract This article is concerned with the Dirichlet eigenvalue problem associated with the ∞ \infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞ \infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for p p -Laplacian in the Euclidean space.

The globalization theorem for the Curvature-Dimension condition

The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X,mathsf {d},{mathfrak {m}}) (so that (text {supp}({mathfrak {m}}),mathsf {d}) is a length-space and {mathfrak {m}}(X) < infty ) verifying the local Curvature-Dimension condition {mathsf {CD}}_{loc}(K,N) with parameters K in {mathbb {R}} and N in (1,infty ), also verifies the global Curvature-Dimension condition {mathsf {CD}}(K,N). In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between L^1- and L^2-optimal-transport–based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.

A metric characterization of Carnot groups

We give a short axiomatic introduction to Carnot groups and their subRiemannian and subFinsler geometry. We explain how such spaces can be metrically described as exactly those proper geodesic spaces that admit dilations and are isometrically homogeneous.

Distance k-sectors exist

The bisector of two nonempty sets P and Q in R d is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k ⩾ 2 is an integer, is a ( k − 1 ) -tuple ( C 1 , C 2 , … , C k − 1 ) such that C i is the bisector of C i − 1 and C i + 1 for every i = 1 , 2 , … , k − 1 , where C 0 = P and C k = Q . This notion, for the case where P and Q are points in R 2 , was introduced by Asano, Matoušek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance 3-sector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension (uniqueness remains open), or more generally, in proper geodesic spaces. The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster–Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.