Riemann curvature tensor on RCD spaces and possible applications

We show that, on every RCD space, it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since, after the works of Petrunin and Zhang–Zhu, we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature – defined in terms of such abstract Riemann tensor – is bounded from below

Second order differentiation formula on RCD∗(K;N) spaces

The aim of this paper is to prove a second order differentiation formula for H2;2 functions along geodesics in RCD∗(K;N) spaces with K ∈R and N < ∞. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that W2-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolations. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: • equiboundedness of densities along entropic interpolations, • local equi-Lipschitz continuity of Schrödinger potentials, • uniform weighted L2 control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the RCD setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, as in the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality

Calculus and Fine Properties of Functions of Bounded Variation on RCD Spaces

We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of RCD spaces. In the infinite dimensional setting, we are able to define an analogue of the distributional differential and, on finite dimensional spaces, we prove fine properties and suitable calculus rules, such as the Vol’pert chain rule for vector valued functions

Local vector measures

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean space. In more general curved contexts, the same objects can be perfectly understood via charts. However, charts are often unavailable in the less regular setting of metric geometry, where still the questions make sense. In this paper we propose a way to deal with this sort of problems and, more generally, to give a meaning to a concept of ‘measure acting in duality with sections of a given bundle’, loosely speaking. Despite the generality, several classical results in measure theory like Riesz's and Alexandrov's theorems have a natural counterpart in this setting. Moreover, as we are going to discuss, the notions introduced here provide a unified framework for several key concepts in nonsmooth analysis that have been introduced more than two decades ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions and Miranda's BV functions. Not surprisingly, the understanding of the structure of these objects improves with the regularity of the underlying space. We are particularly interested in the case of RCD spaces where, as we will argue, the regularity of several key measures of the type we study nicely matches the known regularity theory for vector fields, resulting in a very effective theory. We expect that the notions developed here will help creating stronger links between differential calculus in Alexandrov spaces (based on Perelman's DC charts) and in RCD ones (based on intrinsic tensor calculus)

A Note about the Strong Maximum Principle on RCD Spaces

We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance

Differential structure associated to axiomatic Sobolev spaces

The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (\ue0 la Gol'dshtein\u2013Troyanov) induces \u2013 under suitable locality assumptions \u2013 a first-order differential structure

The Abresch-Gromoll inequality in a non-smooth setting

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian CD(K,N) spaces in the same form as the one available on smooth Riemannian manifolds

Non-collapsed spaces with Ricci curvature bounded from below

\u2014 We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding\u2019s volume convergence theorem and of Cheeger-Colding dimension gap estimate for RCD spaces. In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence

Random laser from engineered nanostructures obtained by surface tension driven lithography

The random laser emission from the functionalized thienyl-S,S-dioxide quinquethiophene (T5OCx) in confined patterns with different shapes is demonstrated. Functional patterning of the light emitter organic material in well defined features is obtained by spontaneous molecular self-assembly guided by surface tension driven (STD) lithography. Such controlled supramolecular nano-aggregates act as scattering centers allowing the fabrication of one-component organic lasers with no external resonator and with desired shape and efficiency. Atomic force microscopy shows that different geometric pattern with different supramolecular organization obtained by the lithographic process tailors the coherent emission properties by controlling the distribution and the size of the random scatterers

Differential of metric valued Sobolev maps

We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is R. We also show compatibility with the concept of co-local weak differential introduced by Convent and Van Schaftingen