4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings

The $L^2$-completion of the space of Riemannian metrics is CAT$(0)$: a shorter proof

We reprove in an easier way a result of Brian Clarke: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the $L^2$-distance is CAT$(0)$. In particular we show that this completion is isometric to the space of $L^2$-maps from a standard probability space to a fixed CAT$(0)$ space.

Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L^2 metric and show that there exists a unique minimal path between any two points. This path is also given explicitly. As an application of these formulas, we show that the metric completion of the manifold of metrics is a CAT(0) space.

Large scale detection of half-flats in CAT(0)-spaces

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0)-spaces, (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds, and (3) a correspondence between metric splittings of a complete, simply connected, non-positively curved Riemannian manifold and the metric splittings of its asympt…

趋向黎曼流形微分学

Towards Riemannian diffeology

本文提出了一种基于切 functor 和度量选项的黎曼微分学框架，并证明了弱黎曼度量诱导的距离性质。

Rank Rigidity for CAT(0) spaces without 3-flats

If a group $Γ$ acts geometrically on a CAT(0) space $X$ without 3-flats, then either $X$ contains a $Γ$-periodic geodesic which does not bound a flat half-plane, or else $X$ is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.

Lusternik-Schnirelmann category and systolic category of low dimensional manifolds

We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns …

Enlargeable Length-structures and Scalar Curvatures

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed $n$-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Ya…

The metric completion of the Riemannian space of Kähler metrics

Let $X$ be a compact Kähler manifold and $\a \in H^{1,1}(X,\R)$ a Kähler class. We study the metric completion of the space $\HH_\a$ of Kähler metrics in $\a$, when endowed with the Mabuchi $L^2$-metric $d$. Using recent ideas of Darvas, we show that the metric completion $(\overline{\HH}_\a,d)$ of $(\HH_\a,d)$ is a CAT(0) space which can be identified with $\E^2(\a)$, a subset of the class $\E^1(\a)$ of positive closed currents with finite energy. We further prove, in the toric setting, that $\overline{\HH}_{\a,tor}=\E_{tor}^2(\a)$.

Completions, branched covers, Artin groups and singularity theory

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(k) metric space for the completion to be CAT(k). We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT(k) if and only if all tangent spaces are CAT(0) and the base has sectional curvature bounded above by k. We also show that the branch…

Existence and uniqueness of optimal transport maps in locally compact $CAT(0)$ spaces

We show that in a locally compact complete $CAT(0)$ space satisfying positive angles property and a disintegration regularity for its canonical Hausdorff measure, there exists a unique optimal transport map that push-forwards a given absolutely continuous probability measure to another probability measure. In particular this holds for the Riemannian manifolds of non-positive sectional curvature and $CAT(0)$ Euclidean polyhedral complexes. Moveover we give a polar factorization result for Borel maps in $CAT(0)$ spaces in terms of optimal transport maps and measure preserving maps.