On energy functionals, Kähler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kähler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kähler potentials on which this inequality holds on higher-dimensional Fano Kähler-Einstein manifolds.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.     

NEW SURFACTANTS FOB SOLUBLE EXTREME PRESSURE CUTTING OILS

The use of nonionic and anionic surfactants is common in soluble cutting oils to facilitate spontaneous emulsification and to keep the formulation stable. Various additives are also introduced in order to achieve high pressure stability, better resistance to high sheers, minimum corrosion and other related properties. Tailor-made surfactants into which specific functional groups are introduced into the hydrophobic chain are of special interest mainlv if extreme pressure (EP) properties can be improved. Those surfactants will help to replace part of the additives and will reduce the cost of the formulation. Incorporation of halogens (chlorine, bromine) to the hydrophobic tail of ethoxylated nonylphenols; sorbitan esters of fatty acids, ethoxylated oleyl alcohols and polyglycerol esters has been carried out. The new surfactants have high specific gravities and therefore can minimize creaming of the emulsion and will improve the EP properties of the soluble cutting oils. Functionalization of the surfactants did not retained the emulsion stability, yet reduced the creaming and the cost of the formulation. It has been also shown that the lubrication properties e.g. the resistance to load and torque have been improved.     

The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge–Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article—that uses tools of convex analysis and can be read independently—we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge–Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.     

Quantization in Geometric Pluripotential Theory

The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite-dimensional spaces of Fubini-Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new Lidskii-type inequality on the space of Kähler metrics, a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, and approximation of finite-energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects. © 2019 Wiley Periodicals, Inc.     

DFT and Empirical Considerations on Electrocatalytic Water/Carbon Dioxide Reduction by CoTMPyP in Neutral Aqueous Solutions**

A combined experimental and density functional theory (DFT) investigation was employed in order to examine the mechanism of electrochemical CO₂ reduction and H₂ formation from water reduction in neutral aqueous solutions. A water soluble cobalt porphyrin, cobalt [5,10,15,20-(tetra-N-methyl-4-pyridyl)porphyrin], (CoTMPyP), was used as catalyst. The possible attachment of different axial ligands as well as their effect on the electrocatalytic cycles were examined. A cobalt porphyrin hydride is a key intermediate which is generated after the initial reduction of the catalyst. The hydride is involved in the formation of H₂ and formate and acts as an indirect proton source for the formation of CO in these H⁺-starving conditions. The experimental results are in agreement with the computations and give new insights into electrocatalytic mechanisms involving water soluble metalloporphyrins. We conclude that in addition to the porphyrin's structure and metal ion center, the electrolyte surroundings play a key role in dictating the products of CO₂/H₂O reduction.     

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L²$L^2$ metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.     

RS-A4 relaxation of flavor and CP violation

I discuss a model based on an A₄ bulk flavor symmetry in the Randall-Sundrum (RS) setup. After discussing the setup and leading order results for the masses and mixings of quarks and leptons, I elaborate on the effect of higher order “cross-talk” corrections, their contributions to flavor violating processes and the resulting constraints on the model parameter space and the Kaluza-Klein (KK) mass scale. In addition, I present a systematic study of higher order corrections to the PMNS matrix in light of the recent measurements of θ ₁₃?>?0 by RENO and Daya Bay. Finally, I also comment on the model new physics contributions to $B_{s,d}\to\mu^{+}\mu^-$ and μ → eγ, in light of the new upper bounds recently set by the LHCb and MEG experiment.