A note on the index of closed minimal hypersurfaces of flat tori

Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients)

Effect of coordinating solvents on the structure of Cu(II)-4,4'-bipyridine coordination polymers

Solvent can play a crucial role in the synthesis of coordination polymers (CPs). Here, this study reports how the coordinating solvent approach (CSA) can be used as an effective tool to control the nature of the final CP. This study exploited the system Cu(II)-4,4 '-bipyridine coupled to different coordinating solvents, such as DMA, DMF and DMSO. This allowed the isolation and structurally characterization of four new CPs: three 2D layered networks and one 1D chain. Moreover, it was evidenced that even adventitious water can play the role of the coordinating solvent in the final CP

The temporal evolution of the Mitu group, south-east Peru – first U-Pb age data

The Eastern Cordillera of southern Peru formed along a crustal zone that has been active as part of the western Gondwana margin since the middle Paleozoic. The present study investigates the Mitu Group of south-east Peru in the area of Abancay-Cusco-Sicuani-Titicaca. This unit comprises continental clastic sediments deposited in syn-sedimentary basins during an extensional period in Permo-Triassic times and has not benefitted from a thorough geochemical-geochronological investigation so far. One of the main reasons for this lack of data is a complex structure of the graben system, tectonically complicated by compressional inversion of the extensional basins during Andean orogeny. Due to dominating coarse-grained clastics, the Mitu Group is devoid of fossils and its age is only poorly bracketed to be Permo -Triassic based on its stratigraphic relation to the underlying Copacabana and overlying Pucara groups. The upper levels of the Copacabana have been constrained by palynology to the Artinskian (Doubinger and Marocco, 1981). However, a hiatus may be observed between the Copacabana and the Mitu groups in most places, rendering the age estimate of the basal Mitu imprecise. The Pucara Group, regarded by Rosas et al. (2007) as thermal sag after Mitu extension, is attributed to the late Triassic - early Jurassic on the basis of ammonite fossils and U-Pb zircon ages from ash beds (Schaltegger et al., 2008). The aim of this study is to provide more accurate and precise age constraints for the age and duration of the Mitu Group by using U-Pb geochronology of volcanic zircon in rhyolitic lavas, and of detrital zircon in clastic sediments. For andesitic volcanic lithologies, age approximations will be obtained by Ar-Ar techniques applied to amphibole and groundmass samples. Field data were obtained from a long and apparently complete section through the Mitu, situated 120km SE of Cusco near the city of Sicuani. This section consists of typical Mitu deposits; continental red beds, breccias and andesitic lavas. However, a zircon-bearing rhyolitic lava at the bottom gives us the opportunity to date the start of Mitu sedimentation by U-Pb ID-TIMS; this analysis will provide a precise age for the base of the Mitu group for the first time. In the Sicuani area the Mitu unconformably overlies the Ambo group, suggesting that the entire Copacabana is missing. Laser-ablation ICP-MS U-Pb data of detrial zircons from a sandstone just below the unconformity indicate a maximum age of latest Carboniferous (303Ma) for the underlying Ambo group. This maximum age overlaps with the palynological age of the lower Copacabana (Azcuy et al., 2002), raising the question whether the Ambo and Copacabana are truly diachronous or just coeval units of different sedimentary facies associations. In another section, 100km W of Cusco, near the city of Abancay, we found Mitu sediments overlying the Copacabana Group. Here the Copacabana contains well preserved plant fossils of the lycopsids family also found elsewhere in Peru and Bolivia. Lack of acidic volcanism during Mitu extension in this region prevents from dating of lavas using the U-Pb method. The detrital zircon population in a sandstone in the lowermost part of the Mitu was analysed for U-Pb ages, using LA-ICP-MS techniques. The youngest zircons in the population are around 235 Ma hence providing a maximum age for the onset of Mitu group sedimentation. The Artinskian age for the upper Copacabana from Doubinger and Marocco (1981) has also been obtained from the Abancay region, establishing a hiatus of some 50 Myrs between the two units. The Mitu Group is intruded by a 220 Ma granite body (Lipa and Saraiva, 2008) indicating significant burial of the sediments at this time. 500km SE of Cusco, on the Bolivian shores of lake Titicaca, the Ambo Group features plant fossils of the Lycopsids family like those found in the Copacabana near Abancay. Our detrital zircon LA-ICPMS study on a quartz arenite just below the fossils indicates a maximum U-Pb age of 343Ma. However a zircon-bearing ash bed will allow for more precise calibration of the fossil age by ID-TIMS techniques. The zircon U-Pb data will provide a test whether the Copacabana and the Ambo group are indeed diachronous or just lateral variations of a sedimentary system

Compactness analysis for free boundary minimal hypersurfaces

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control

Compactness of the Space of Minimal Hypersurfaces with Bounded Volume and p-th Jacobi Eigenvalue

Given a closed Riemannian manifold of dimension less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p≥2 one gains strong convergence to a smooth limit submanifold away from at most p−1 points

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in $\mathbb{R}^3$ unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified

An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.Comment: some references adde

Index estimates for free boundary minimal hypersurfaces

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser–Schoen and Folha–Pacard–Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed

Current trends on subtotal petrosectomy with cochlear implantation in recalcitrant chronic middle ear disorders

Objective. To establish the safety and effectiveness of subtotal petrosectomy with cochlear implantation in patients affected by chronic middle ear disorders to refractory to previous surgical treatments. Methods. A multicentre, retrospective study was conducted on patients affected by recalcitrant chronic middle ear disorders who underwent cochlear implantation in combi-nation with subtotal petrosectomy. Patients’ details were collected from databases of 11 Italian tertiary referral centres. Additionally, a review of the most updated literature was carried out. Results. 55 patients were included with a mean follow-up time of 44 months. Cholestea-toma was the most common middle ear recurrent pathology and 50.9% of patients had an open cavity. 80% of patients underwent a single stage surgery. One case of explantation for device failure was reported among the 7 patients with post-operative complications. Conclusions. Subtotal petrosectomy with cochlear implantation is a benchmark for management of patients with recalcitrant chronic middle ear disorders. A single stage procedure is the most recommended strategy. Optimal follow-up is still debated. Further studies are required to investigate the role of this surgery in paediatric patients