Volume 2, Issue 1

Let A = (a ij ) ∊ M n (ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, d X ), let γ(A, d p x ) be the infimum over those γ ∊ (0,∞] for which every x 1 , . . . , x n ∊ X satisfy Thus γ (A, d p x ) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel d p X : X × X →[0,∞). We study pairs of metric spaces (X, d X ) and (Y, d Y ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, d p X ) ≤Ψ (A, d p Y ) for every symmetric stochastic A ∊ Mn(ℝ) with (A, d p Y ) < ∞. When Ψ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if n ∊ ℕ and p ∊ (2,∞) then for every f 1 , . . . , f n ∊ L p there exist x 1 , . . . , x n ∊ L 2 such that and This statement is impossible for p ∊ [1, 2), and the asymptotic dependence on p in (0.1) is sharp. We also obtain the best known lower bound on the L p distortion of Ramanujan graphs, improving over the work of Matoušek. Links to Bourgain-Milman-Wolfson type and a conjectural nonlinear Maurey-Pisier theorem are studied.

We develop the boundary theory of rough CAT(0) spaces, a class of length spaces that contains both Gromov hyperbolic length spaces and CAT(0) spaces. The resulting theory generalizes the common features of the Gromov boundary of a Gromov hyperbolic length space and the ideal boundary of a complete CAT(0) space. It is not assumed that the spaces are geodesic or proper

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.

We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.

In this paperwe give new existence results for complete non-orientable minimal surfaces in ℝ 3 with prescribed topology and asymptotic behavior

We present inversion results for Lipschitz maps f : Ω ⊂ ℝ N → ( Y, d ) and stability of inversionfor uniformly convergent sequences. These results are based on the Area Formula and on the l.s.c. of metricJacobians.

We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J 2 -condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.

A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra.We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to bethe natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meantto stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensionalintrinsic Lipschitz graphs are sets with locally finite G-perimeter. From this a Rademacher’s typetheorem for one codimensional graphs in a general class of groups is proved.

In this paper we consider compact, Riemannian manifolds M 1 , M 2 each equipped with a oneparameterfamily of metrics g 1 (t), g 2 (t) satisfying the Ricci flow equation. Adopting the characterization ofsuper-solutions to the Ricci flow developed by McCann-Topping, we define a super Ricci flow for a family ofdistance metrics defined on the disjoint union M 1 ⊔ M 2 . In particular, we show such a super Ricci flow propertyholds provided the distance function between points in M1 and M2 is itself a super solution of the heatequation on M 1 × M 2 . We also discuss possible applications and examples.

This paper is a study of harmonic maps fromRiemannian polyhedra to locally non-positively curvedgeodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functionsand harmonic maps under two different assumptions on the source space. First we prove the analogue ofthe Schoen-Yau Theorem on a complete pseudomanifolds with non-negative Ricci curvature. Then we study2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.

We consider (bounded) Besicovitch sets in the Heisenberg group and prove that L p estimates forthe Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.

Let be an open half-space or slab in ℝ n+1 endowed with a perturbation of the Gaussian measureof the form f (p) := exp(ω(p) − c|p| 2 ), where c > 0 and ω is a smooth concave function depending only onthe signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω uniquely minimize the weighted perimeter in Ω among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spacesparallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity.Our methods also apply for = ℝ n+1 , which produces in particular the classification of stable sets in Gaussspace and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to studythe weighted minimizers when the perturbation term ω is concave and possibly non-smooth.

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit ageneralized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simpleproof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology, called coconvextopology, agrees with the usually weak topology in Banach spaces. An example of a CAT(0)-spacewith weak topology which is not Hausdorff is given.In the end existence and uniqueness of generalized barycenters is shown, an application to isometric groupactions is given and a Banach-Saks property is proved.