Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Structure Theory of Metric-measure Spaces with Lower Ricci Curvature Bounds I

We prove that a metric measure space (X, d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting ...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure

In prior work [4] of the first two authors with Savaré, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X, d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In [4] the RCD(K,∞) property is defined in thr...

Sharp and Rigid Isoperimetric Inequalities in Metric-measure Spaces with Lower Ricci Curvature Bounds

We prove that if (X, d,m) is a metric measure space with m(X) = 1 having (in a synthetic sense) Ricci curvature bounded from below by K > 0 and dimension bounded above by N ∈ [1,∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any K ∈ R, N ≥ 1 and upper diameter bounds) hold, i.e. the isoper...