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

In a prior work of the first two authors with Savar´e, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces (X, d, m) was introduced, and the corresponding class of spaces was 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 this prior work the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In the above-mentioned work only finite reference measure…

Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces

We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.

Collagen-low molecular weight hyaluronic acid semi-interpenetrating network loaded with gelatin microspheres for cell and growth factor delivery for nucleus pulposus regeneration.

Intervertebral disc (IVD) degeneration is one of the main causes of low back pain. Current surgical treatments are complex and generally do not fully restore spine mobility. Development of injectable extracellular matrix-based hydrogels offers an opportunity for minimally invasive treatment of IVD degeneration. Here we analyze a specific formulation of collagen-low molecular weight hyaluronic acid (LMW HA) semi-interpenetrating network (semi-IPN) loaded with gelatin microspheres as a potential material for tissue engineering of the inner part of the IVD, the nucleus pulposus (NP). The material displayed a gel-like behavior, it was easily injectable as demonstrated by suitable tests and did …

Optimal C∞-approximation of functions with exponentially or sub-exponentially integrable derivative