38 research outputs found
Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval
In this essay, we discuss the notion of optimal transport on geodesic measure
spaces and the associated (2-)Wasserstein distance. We then examine
displacement convexity of the entropy functional on the space of probability
measures. In particular, we give a detailed proof that the Lott-Villani-Sturm
notion of generalized Ricci bounds agree with the classical notion on smooth
manifolds. We also give the proof that generalized Ricci bounds are preserved
under Gromov-Hausdorff convergence. In particular, we examine in detail the
space of probability measures over the interval, equipped with the
Wasserstein metric . We show that this metric space is isometric to a
totally convex subset of a Hilbert space, , which allows for concrete
calculations, contrary to the usual state of affairs in the theory of optimal
transport. We prove explicitly that has vanishing Alexandrov
curvature, and give an easy to work with expression for the entropy functional
on this space. In addition, we examine finite dimensional Gromov-Hausdorff
approximations to this space, and use these to construct a measure on the limit
space, the entropic measure first considered by Von Renesse and Sturm. We
examine properties of the measure, in particular explaining why one would
expect it to have generalized Ricci lower bounds. We then show that this is in
fact not true. We also discuss the possibility and consequences of finding a
different measure which does admit generalized Ricci lower bounds.Comment: 47 pages, 9 figure
Large isoperimetric regions in asymptotically hyperbolic manifolds
We show the existence of isoperimetric regions of sufficiently large volumes
in general asymptotically hyperbolic three manifolds. Furthermore, we show that
large coordinate spheres in compact perturbations of
Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the
context of the asymptotically hyperbolic Penrose inequality.
Our results require that the scalar curvature of the metric satisfies
, and we construct an example of a compact perturbation of
Schwarzschild-anti-deSitter without so that large centered
coordinate spheres are not isoperimetric. The necessity of scalar curvature
bounds is in contrast with the analogous uniqueness result proven by Bray for
compact perturbations of Schwarzschild, where no such scalar curvature
assumption is required.
This demonstrates that from the point of view of the isoperimetric problem,
mass behaves quite differently in the asymptotically hyperbolic setting
compared to the asymptotically flat setting. In particular, in the
asymptotically hyperbolic setting, there is an additional quantity, the
"renormalized volume," which has a strong effect on the large-scale geometry of
volume.Comment: 57 pages, 1 figure. Comments welcome
Uniqueness of asymptotically conical tangent flows
Singularities of the mean curvature flow of an embedded surface in R^3 are
expected to be modelled on self-shrinkers that are compact, cylindrical, or
asymptotically conical. In order to understand the flow before and after the
singular time, it is crucial to know the uniqueness of tangent flows at the
singularity.
In all dimensions, assuming the singularity is multiplicity one, uniqueness
in the compact case has been established by the second-named author, and in the
cylindrical case by Colding-Minicozzi. We show here the uniqueness of
multiplicity-one asymptotically conical tangent flows for mean curvature flow
of hypersurfaces.
In particular, this implies that when a mean curvature flow has a
multiplicity-one conical singularity model, the evolving surface at the
singular time has an (isolated) regular conical singularity at the singular
point. This should lead to a complete understanding of how to "flow through"
such a singularity.Comment: 40 page
Effective versions of the positive mass theorem
The study of stable minimal surfaces in Riemannian -manifolds
with non-negative scalar curvature has a rich history. In this paper, we prove
rigidity of such surfaces when is asymptotically flat and has horizon
boundary. As a consequence, we obtain an effective version of the positive mass
theorem in terms of isoperimetric or, more generally, closed volume-preserving
stable CMC surfaces that is appealing from both a physical and a purely
geometric point of view. We also include a proof of the following conjecture of
R. Schoen: An asymptotically flat Riemannian -manifold with non-negative
scalar curvature that contains an unbounded area-minimizing surface is
isometric to flat .Comment: All comments welcome! The final version has appeared in Invent. Mat