arXiv subjects

Robert J McCann

Publications and source records attributed to Robert J McCann.

5 recordsLinked to original sources

arXiv2023

Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. This article initiates the development of an analogous theory for lower Ricci curvature bounds in timelike directions on a (globally hyperbolic) Lorentzian manifold. In particular, we lift fractional powers of the Lorentz distance (a.k.a. time separation function) to probability measures on spacetime, and show the strong energy condition of Hawking and Penrose is equivalent to geodesic convexity of the Boltzmann-Shannon entropy there. This represents a significant first step towards a formulation of the strong energy condition and exploration of its consequences in nonsmooth spacetimes, and hints at new connections linking the theory of gravity to the second law of thermodynamics.

math-ph↗

arXiv2021

Isodiametry, variance, and regular simplices from particle interactions

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained -- apart from translations and rotations -- by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show these configurations are at least local energy minimizers in the relevant $d_\infty$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based on a simple isodiametric variance bound which characterizes regular simplices: it shows that among probability measures on ${\mathbf R}^n$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter) regular simplex.

math-ph↗

arXiv2019

Optimal transportation between unequal dimensions

We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,α}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works.

math.AP↗

arXiv2016

Multi- to one-dimensional transportation

Fix probability densities $f$ and $g$ on open sets $X \subset \mathbf{R}^m$ and $Y \subset \mathbf{R}^n$ with $m\ge n\ge1$. Consider transporting $f$ onto $g$ so as to minimize the cost $-s(x,y)$. We give a non-degeneracy condition (a) on $s \in C^{1,1}$ which ensures the set of $x$ paired with [$g$-a.e.] $y\in Y$ lie in a codimension $n$ submanifold of $X$. Specializing to the case $m>n=1$, we discover a nestedness criteria relating $s$ to $(f,g)$ which allows us to construct a unique optimal solution in the form of a map $F:X \longrightarrow \overline Y$. When $s\in C^2 \cap W^{3,1}$ and $\log f$ and $\log g$ are bounded, the Kantorovich dual potentials $(u,v)$ satisfy $v \in C^{1,1}_{loc}(Y)$, and the normal velocity $V$ of $F^{-1}(y)$ with respect to changes in $y$ is given by $V(x) = v"(f(x))-s_{yy}(x,f(x))$. Positivity (b) of $V$ locally implies a Lipschitz bound on $f$; moreover, $v \in C^2$ if ${F^{-1}(y)}$ intersects $\partial X \in C^1$ transversally (c). On subsets where (a)-(c) can be be quantified, for each integer $r \ge1$ the norms of $u,v \in C^{r+1,1}$ and $F \in C^{r,1}$ are controlled by these bounds, $||\log f,\log g, \partial X ||_{C^{r-1,1}}, ||\partial X||_{C^{1,1}}$, $||s||_{C^{r+1,1}}$, and the smallness of $F^{-1}(y)$. We give examples showing regularity extends from $X$ to part of $\bar X$, but not from $Y$ to $\bar Y$. We also show that when $s$ remains nested for all $(f,g)$, the problem in $\mathbf{R}^m \times \mathbf{R}$ reduces to a supermodular problem in $\mathbf{R} \times \mathbf{R}$.

math.AP↗

arXiv2012

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

math.OC↗