Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

Abstract deformations of the CR structure of a compact strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ are encoded by complex functions on $M$. In sharp contrast with the higher dimensional case, the natural integrability condition for $3$-dimensional CR structures is vacuous, and generic deformations of a compact strictly pseudoconvex hypersurface $M\subseteq \mathbb{C}^2$ are not embeddable even in $\mathbb{C}^N$ for any $N$. A fundamental (and difficult) problem is to characterize when a complex function on $M \subseteq \mathbb{C}^2$ gives rise to an actual deformation of $M$ inside $\mathbb{C}^2$. In this paper we study the embeddability of families of deformations of a given embedded CR $3$-manifold, and the structure of the space of embeddable CR structures on $S^3$. We show that the space of embeddable deformations of the standard CR $3$-sphere is a Frechet submanifold of $C^{\infty}(S^3,\mathbb{C})$ near the origin. We establish a modified version of the Cheng-Lee slice theorem in which we are able to characterize precisely the embeddable deformations in the slice (in terms of spherical harmonics). We also introduce a canonical family of embeddable deformations and corresponding embeddings starting with any infinitesimally embeddable deformation of the unit sphere in $\mathbb{C}^2$.Comment: 42 page

Conformal submanifolds, distinguished submanifolds, and integrability

For conformal geometries of Riemannian signature, we provide a comprehensive and explicit treatment of the core local theory for embedded submanifolds of arbitrary dimension. This is based in the conformal tractor calculus and includes a conformally invariant Gauss formula leading to conformal versions of the Gauss, Codazzi, and Ricci equations. It provides the tools for proliferating submanifold conformal invariants, as well for extending to conformally singular Riemannian manifolds the notions of mean curvature and of minimal and CMC submanifolds. A notion of distinguished submanifold is defined by asking the tractor second fundamental form to vanish. We show that for the case of curves this exactly characterises conformal geodesics (a.k.a. conformal circles) while for hypersurfaces it is the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that in all dimensions this coincides with the submanifold being weakly conformally circular, meaning that ambient conformal circles remain in the submanifold. Stronger notions of conformal circularity are then characterised similarly. Next we provide a very general theory and construction of quantities that are necessarily conserved along distinguished submanifolds. This first integral theory thus vastly generalises the results available for conformal circles in [56]. We prove that any normal solution to an equation from the class of first BGG equations can yield such a conserved quantity, and show that it is easy to provide explicit formulae for these. Finally we prove that the property of being distinguished is also captured by a type of moving incidence relation. This second characterisation is used to show that, for suitable solutions of conformal Killing-Yano equations, a certain zero locus of the solution is necessarily a distinguished submanifold.Comment: 87 page

On the Lichnerowicz conjecture for CR manifolds with mixed signature

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1,n-1)$ which is not locally CR flat and admits an essential CR vector fields. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.Comment: 7 page

Unified treatment algorithm for the management of crotaline snakebite in the United States: results of an evidence-informed consensus workshop

<p>Abstract</p> <p>Background</p> <p>Envenomation by crotaline snakes (rattlesnake, cottonmouth, copperhead) is a complex, potentially lethal condition affecting thousands of people in the United States each year. Treatment of crotaline envenomation is not standardized, and significant variation in practice exists.</p> <p>Methods</p> <p>A geographically diverse panel of experts was convened for the purpose of deriving an evidence-informed unified treatment algorithm. Research staff analyzed the extant medical literature and performed targeted analyses of existing databases to inform specific clinical decisions. A trained external facilitator used modified Delphi and structured consensus methodology to achieve consensus on the final treatment algorithm.</p> <p>Results</p> <p>A unified treatment algorithm was produced and endorsed by all nine expert panel members. This algorithm provides guidance about clinical and laboratory observations, indications for and dosing of antivenom, adjunctive therapies, post-stabilization care, and management of complications from envenomation and therapy.</p> <p>Conclusions</p> <p>Clinical manifestations and ideal treatment of crotaline snakebite differ greatly, and can result in severe complications. Using a modified Delphi method, we provide evidence-informed treatment guidelines in an attempt to reduce variation in care and possibly improve clinical outcomes.</p

Variation in Structure and Process of Care in Traumatic Brain Injury: Provider Profiles of European Neurotrauma Centers Participating in the CENTER-TBI Study.

INTRODUCTION: The strength of evidence underpinning care and treatment recommendations in traumatic brain injury (TBI) is low. Comparative effectiveness research (CER) has been proposed as a framework to provide evidence for optimal care for TBI patients. The first step in CER is to map the existing variation. The aim of current study is to quantify variation in general structural and process characteristics among centers participating in the Collaborative European NeuroTrauma Effectiveness Research in Traumatic Brain Injury (CENTER-TBI) study. METHODS: We designed a set of 11 provider profiling questionnaires with 321 questions about various aspects of TBI care, chosen based on literature and expert opinion. After pilot testing, questionnaires were disseminated to 71 centers from 20 countries participating in the CENTER-TBI study. Reliability of questionnaires was estimated by calculating a concordance rate among 5% duplicate questions. RESULTS: All 71 centers completed the questionnaires. Median concordance rate among duplicate questions was 0.85. The majority of centers were academic hospitals (n = 65, 92%), designated as a level I trauma center (n = 48, 68%) and situated in an urban location (n = 70, 99%). The availability of facilities for neuro-trauma care varied across centers; e.g. 40 (57%) had a dedicated neuro-intensive care unit (ICU), 36 (51%) had an in-hospital rehabilitation unit and the organization of the ICU was closed in 64% (n = 45) of the centers. In addition, we found wide variation in processes of care, such as the ICU admission policy and intracranial pressure monitoring policy among centers. CONCLUSION: Even among high-volume, specialized neurotrauma centers there is substantial variation in structures and processes of TBI care. This variation provides an opportunity to study effectiveness of specific aspects of TBI care and to identify best practices with CER approaches

Risk of SARS-CoV-2 reinfection during multiple Omicron variant waves in the UK general population

SARS-CoV-2 reinfections increased substantially after Omicron variants emerged. Large-scale community-based comparisons across multiple Omicron waves of reinfection characteristics, risk factors, and protection afforded by previous infection and vaccination, are limited. Here we studied ~45,000 reinfections from the UK’s national COVID-19 Infection Survey and quantified the risk of reinfection in multiple waves, including those driven by BA.1, BA.2, BA.4/5, and BQ.1/CH.1.1/XBB.1.5 variants. Reinfections were associated with lower viral load and lower percentages of self-reporting symptoms compared with first infections. Across multiple Omicron waves, estimated protection against reinfection was significantly higher in those previously infected with more recent than earlier variants, even at the same time from previous infection. Estimated protection against Omicron reinfections decreased over time from the most recent infection if this was the previous or penultimate variant (generally within the preceding year). Those 14–180 days after receiving their most recent vaccination had a lower risk of reinfection than those &gt;180 days from their most recent vaccination. Reinfection risk was independently higher in those aged 30–45 years, and with either low or high viral load in their most recent previous infection. Overall, the risk of Omicron reinfection is high, but with lower severity than first infections; both viral evolution and waning immunity are independently associated with reinfection

Antibody responses and correlates of protection in the general population after two doses of the ChAdOx1 or BNT162b2 vaccines

Antibody responses are an important part of immunity after Coronavirus Disease 2019 (COVID-19) vaccination. However, antibody trajectories and the associated duration of protection after a second vaccine dose remain unclear. In this study, we investigated anti-spike IgG antibody responses and correlates of protection after second doses of ChAdOx1 or BNT162b2 vaccines for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in the United Kingdom general population. In 222,493 individuals, we found significant boosting of anti-spike IgG by the second doses of both vaccines in all ages and using different dosing intervals, including the 3-week interval for BNT162b2. After second vaccination, BNT162b2 generated higher peak levels than ChAdOX1. Older individuals and males had lower peak levels with BNT162b2 but not ChAdOx1, whereas declines were similar across ages and sexes with ChAdOX1 or BNT162b2. Prior infection significantly increased antibody peak level and half-life with both vaccines. Anti-spike IgG levels were associated with protection from infection after vaccination and, to an even greater degree, after prior infection. At least 67% protection against infection was estimated to last for 2–3 months after two ChAdOx1 doses, for 5–8 months after two BNT162b2 doses in those without prior infection and for 1–2 years for those unvaccinated after natural infection. A third booster dose might be needed, prioritized to ChAdOx1 recipients and those more clinically vulnerable