Quasicircles and width of Jordan curves in CP1

We study a notion of ‘width’ for Jordan curves in (Formula presented.), paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti-de Sitter geometry was used by Bonsante–Schlenker to characterize quasicircles among a larger class of Jordan curves in the boundary of anti de Sitter space. In contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometry

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature K 2 Π1; 0/ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of K, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti-de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti-de Sitter spacetimes

Some open questions on anti-de Sitter geometry

We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open questions. 19 page

Norm estimates of complex symmetric operators applied to quantum systems

This paper communicates recent results in theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schr\"odinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schr\"odinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schr\"odinger operators appearing in the complex scaling theory of resonances

Candidate Genes for Chromosomes 6 and 10: Quantitative Trait Loci for Age-Related Retinal Degeneration in Mice

Purpose: In a previous study, several quantitative trait loci (QTL) that influence age-related degeneration (ageRD) were identified in a cross between the albino strains B6(Cg)-Tyr(c-2J)/J (B6a) and BALB/cByJ (C). The Chromosome (Chr) 6 and Chr 10 QTL were the strongest and most highly significant loci and both involved B6a protective alleles. The QTL were responsible for 21% and 9% of the variance in phenotypes, respectively. We focused on these two QTL to identify candidate genes. Methods: DNA microarrays were used for the two mouse strains at four and eight months of age to identify genes that are differentially regulated and map to either QTL. Gene Ontology (GO) analysis of the differentially expressed genes was performed to identify possible processes and pathways associated with ageRD. To identify additional candidates, database analyses (Positional Medline or PosMed) were used. Based on differential expression, PosMed, and the presence of reported polymorphisms, five genes per QTL were selected for further study by sequencing analysis and qRT-PCR. Tumor necrosis factor, alpha-induced protein 3 (Tnfaip3; on a C57BL/6J (B6) background) was phenotypically tested. Single nucleotide polymorphisms (SNPs) flanking this gene were correlated with outer nuclear layer thickness (ONL), and eight-month-old Tnfaip3(+/-) mice were tested for ageRD. Results: Polymorphisms were found in the coding regions of eight genes. Changes in gene expression were identified by qRT-PCR for Hexokinase 2 (Hk2) and Docking protein 1 (Dok1) at four months and for Dok1 and Tnfaip3 at eight months. Tnfaip3 was selected for phenotypic testing due to differential expression and the presence of two nonsynonymous mutations. However, when ONL thickness was compared in eight-month-old congenic Tnfaip3(+/-) and Tnfaip3(+/+) mice, no differences were found, suggesting that Tnfaip3 is not the quantitative trait gene (QTG) for the Chr 10 QTL. The GO analysis revealed that GO terms associated with stress and cell remodeling are overrepresented in the ageRD-sensitive C strain compared with the B6a strain with age (eight months). In the ageRD-resistant B6a strain, compared with the C strain, GO terms associated with antioxidant response and the regulation of blood vessel size are overrepresented with age. Conclusions: The analyses of differentially expressed genes and the PosMed database yielded candidate genes for the Chr 6 and Chr 10 QTL. HtrA serine peptidase 2 (Htra2), Dok1, and Tnfaip3 were deemed most promising because of their known roles in apoptosis and our finding of nonsynonymous substitutions between B6a and C strains. While Tnfaip3 was excluded as the QTG for the Chr 10 QTL, Dok1 and Htra2 remain good candidates for the Chr 6 QTL. Finally, the GO term analysis further supports the general hypothesis that oxidative stress is involved in ageRD

Cone rod dystrophies

Cone rod dystrophies (CRDs) (prevalence 1/40,000) are inherited retinal dystrophies that belong to the group of pigmentary retinopathies. CRDs are characterized by retinal pigment deposits visible on fundus examination, predominantly localized to the macular region. In contrast to typical retinitis pigmentosa (RP), also called the rod cone dystrophies (RCDs) resulting from the primary loss in rod photoreceptors and later followed by the secondary loss in cone photoreceptors, CRDs reflect the opposite sequence of events. CRD is characterized by primary cone involvement, or, sometimes, by concomitant loss of both cones and rods that explains the predominant symptoms of CRDs: decreased visual acuity, color vision defects, photoaversion and decreased sensitivity in the central visual field, later followed by progressive loss in peripheral vision and night blindness. The clinical course of CRDs is generally more severe and rapid than that of RCDs, leading to earlier legal blindness and disability. At end stage, however, CRDs do not differ from RCDs. CRDs are most frequently non syndromic, but they may also be part of several syndromes, such as Bardet Biedl syndrome and Spinocerebellar Ataxia Type 7 (SCA7). Non syndromic CRDs are genetically heterogeneous (ten cloned genes and three loci have been identified so far). The four major causative genes involved in the pathogenesis of CRDs are ABCA4 (which causes Stargardt disease and also 30 to 60% of autosomal recessive CRDs), CRX and GUCY2D (which are responsible for many reported cases of autosomal dominant CRDs), and RPGR (which causes about 2/3 of X-linked RP and also an undetermined percentage of X-linked CRDs). It is likely that highly deleterious mutations in genes that otherwise cause RP or macular dystrophy may also lead to CRDs. The diagnosis of CRDs is based on clinical history, fundus examination and electroretinogram. Molecular diagnosis can be made for some genes, genetic counseling is always advised. Currently, there is no therapy that stops the evolution of the disease or restores the vision, and the visual prognosis is poor. Management aims at slowing down the degenerative process, treating the complications and helping patients to cope with the social and psychological impact of blindness

Loss of the Metalloprotease ADAM9 Leads to Cone-Rod Dystrophy in Humans and Retinal Degeneration in Mice

Cone-rod dystrophy (CRD) is an inherited progressive retinal dystrophy affecting the function of cone and rod photoreceptors. By autozygosity mapping, we identified null mutations in the ADAM metallopeptidase domain 9 (ADAM9) gene in four consanguineous families with recessively inherited early-onset CRD. We also found reduced photoreceptor responses in Adam9 knockout mice, previously reported to be asymptomatic. In 12-month-old knockout mice, photoreceptors appear normal, but the apical processes of the retinal pigment epithelium (RPE) cells are disorganized and contact between photoreceptor outer segments (POSs) and the RPE apical surface is compromised. In 20-month-old mice, there is clear evidence of progressive retinal degeneration with disorganized POS and thinning of the outer nuclear layer (ONL) in addition to the anomaly at the POS-RPE junction. RPE basal deposits and macrophages were also apparent in older mice. These findings therefore not only identify ADAM9 as a CRD gene but also identify a form of pathology wherein retinal disease first manifests at the POS-RPE junction