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

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

Quasi-Continuous Vector Fields on RCD Spaces

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined m-a.e. In this paper we introduce the concept of tensor field defined \u20182-capacity-a.e.\u2019 and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative

Experimental evidence of replica symmetry breaking in random lasers

Spin-glass theory is one of the leading paradigms of complex physics and describes condensed matter, neural networks and biological systems, ultracold atoms, random photonics, and many other research fields. According to this theory, identical systems under identical conditions may reach different states and provide different values for observable quantities. This effect is known as Replica Symmetry Breaking and is revealed by the shape of the probability distribution function of an order parameter named the Parisi overlap. However, a direct experimental evidence in any field of research is still missing. Here we investigate pulse-to-pulse fluctuations in random lasers, we introduce and measure the analogue of the Parisi overlap in independent experimental realizations of the same disordered sample, and we find that the distribution function yields evidence of a transition to a glassy light phase compatible with a replica symmetry breaking.Comment: 10 pages, 5 figure