Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand's theorem, Kaufman's theorem, and Falconer's theorem in the above geometrical settings.

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

The effect of projections on dimension in the Heisenberg group

We prove analogs of classical almost sure dimension theorems for Euclidean projection mappings in the first Heisenberg group, equipped with a sub-Riemannian metric.

Seismic anisotropy in the Morcles nappe shear zone: Implications for seismic imaging of crustal scale shear zones

Microstructures and textures of calcite mylonites from the Morcles nappe large-scale shear zone in southwestern Switzerland develop principally as a function of 1) extrinsic physical parameters including temperature, stress, strain, strain rate and 2) intrinsic parameters, such as mineral composition. We collected rock samples at a single location from this shear zone, on which laboratory ultrasonic velocities, texture and microstructures were investigated and quantified. The samples had different concentration of secondary mineral phases (< 5 up to 40 vol.%). Measured seismic P wave anisotropy ranges from 6.5% for polyphase mylonites (~ 40 vol.%) to 18.4% in mylonites with < 5 vol.% secondary phases. Texture strength of calcite is the main factor governing the seismic P wave anisotropy. Measured S wave splitting is generally highest in the foliation plane, but its origin is more difficult to explain solely by calcite texture. Additional texture measurements were made on calcite mylonites with low concentration of secondary phases (≤ 10 vol.%) along the metamorphic gradient of the shear zone (15 km distance). A systematic increase in texture strength is observed moving from the frontal part of the shear zone (anchimetamorphism; 280 °C) to the higher temperature, basal part (greenschist facies; 350–400 °C). Calculated P wave velocities become increasingly anisotropic towards the high-strain part of the nappe, from an average of 5.8% in the frontal part to 13.2% in the root of the basal part. Secondary phases raise an additional complexity, and may act either to increase or decrease seismic anisotropy of shear zone mylonites. In light of our findings we reinterpret the origin of some seismically reflective layers in the Grône–Zweisimmen line in southwestern Switzerland (PNR20 Swiss National Research Program). We hypothesize that reflections originate in part from the lateral variation in textural and microstructural arrangement of calcite mylonites in shear zones.

Dating polygenetic metamorphic assemblages along a transect across the Western Alps

Multichronometric analyses were performed on samples from a transect in the French-Italian Western Alps crossing nappes derived from the Briançonnais terrane and the Piemonte-Liguria Ocean, in an endeavour to constrain the high-pressure (HP) metamorphism and the retrogression history. 12 samples of white mica were analysed by 39Ar-40Ar stepwise heating, complemented by 2 samples from the Monte Rosa 100 km to the NE and also attributed to the Briançonnais terrane. One Sm-Nd and three Lu-Hf garnet ages from eclogites were also obtained. White mica ages decrease from ca. 300 Ma in the westernmost samples (Zone Houillère), reaching ca. 300 °C during Alpine metamorphism, to < 48 Ma in the internal units to the East, which reached ca. 500 °C during Alpine orogeny. The conventional “thermochronological” interpretation postulates Cretaceous Eo-Alpine HP metamorphism and younger “cooling ages” in the higher-temperature samples. However, Eocene Lu-Hf and Sm-Nd ages from the same samples cannot be interpreted as post-metamorphic cooling ages, which makes a Cretaceous eclogitization untenable. The age date from this transect require instead to replace conventional “thermochronology” by an approach combining age dating with detailed geochemical, petrological and microstructural investigations. Petrology reveals important mineralogical differences along the transect. Samples from the Zone Houillère mostly contain detrital mica. White mica with Si > 6.45 atoms per formula unit becomes more abundant eastward. Across the whole traverse, HP phengitic mica forms the D1 foliation. Syn-D2 mica is Si-poorer and associated with nappe stacking, exhumation, and hydrous retrogression under greenschist facies conditions. D1 phengite is very often corroded, overgrown or intergrown by syn-D2 muscovite. Most importantly, syn-D2 recrystallization is not limited to S2 schistosity domains; microchemical fingerprinting shows that it also can form pseudomorphs after crystals that could be mistaken to have formed during D1 based on microstructural arguments alone. Thereby the Cl concentration in white mica is a useful discriminator, since D2 retrogression was associated with a less saline fluid than eclogitization. Once the petrological stage is set, geochronology is straightforward. All samples contain mixtures of detrital, syn-D1 and syn-D2 mica, and retrogression phases (D3) in greatly varying proportions according to local pressure-temperature-fluid activity-deformation conditions. The correlation of age vs. Cl/K clearly identifies 47 ± 1 Ma as the age of formation of syn-D1 mica along the entire transect, including the Monte Rosa nappe samples. The inferred age of the greenschist-facies low-Si syn-D2 mica generation ranges within 39-43 Ma, with local variations. Coexistence of D1 and D2 ages, and the constancy of non-reset D1 ages along the entire transect, are strong evidence that the D1 white mica ages are very close to formation ages. Volume diffusion of Ar in white mica (activation energy E = 250 kJ/mol; pressure-adjusted diffusion coefficient D’0 < 0.03 cm2 s-1) has a subordinate effect on mineral ages compared to both prograde and retrograde recrystallization in most samples. Eocene Lu-Hf and Sm-Nd garnet ages are prograde and predate the HP peak.

Remarks on Lipschitz domains in Carnot groups

In this Note we present the basic features of the theory of Lipschitz maps within Carnot groups as it is developed in [8], and we prove that intrinsic Lipschitz domains in Carnot groups are uniform domains.

Age of cleft monazites in the eastern Tauern Window: constraints on crystallization conditions of hydrothermal monazite

Monazite-bearing Alpine clefts located in the Sonnblick region of the eastern Tauern Window, Austria, are oriented perpendicular to the foliation and lineation. Ion probe (SIMS) Th–Pb and U–Pb dating of four cleft monazites yields crystallization ages of different growth domains and aggregate regions ranging from 18.99 ± 0.51 to 15.00 ± 0.51 Ma. The crystallization ages obtained are overlapping or slightly younger than zircon fission track ages but older than zircon (U–Th)/He cooling ages from the same area. This constrains cleft monazite crystallization in this area to *300–200 �C. LA-ICP-MS data of dated hydrothermal monazites indicate that in graphite-bearing, reduced host lithologies, cleft monazite is poor in As and has higher La/Yb values and U concentrations, whereas in oxidised host rocks opposite trends are observed. Monazites show negative Eu anomalies and variable La/Yb values ranging from 520 to 6050. The positive correlation between Ca and Sr concentration indicates dissolution of plagioclase or carbonates as the source of these elements. The data show that early exhumation and cleft formation in the Tauern is related to metamorphic dome formation caused by the collision of the Adriatic with the European plate and that monazite crystallization in the clefts occurred later. Our data also demonstrate that hydrothermal monazite ages offer great potential in helping to constrain the chronology of exhumation in collisional orogens.

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Steiner’s formula in the Heisenberg group

Steiner’s tube formula states that the volume of an ϵ-neighborhood of a smooth regular domain in Rn is a polynomial of degree n in the variable ϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.

Age constraints on the tectonic evolution of the Itremo region in Central Madagascar

The Itremo region in Central Madagascar comprises a deformed metasedimentary sequence (Itremo Group) that has undergone greenschist to lower amphibolite facies metamorphism. During a first phase of deformation (D1) Itremo Group sediments were deformed into a fold-and-thrust belt and transported toward the E to NE on top of migmatitic gneisses rocks of Anatananarivo block. A second phase of deformation (D2) affected both the fold-and-thrust belt and structurally underlying units, and formed large-scale N-S trending folds with steeply dipping axial planes. A Late Neoproterozoic Th–U–Pb XRF monazite age (565±17 Ma) dates the emplacement of a granite that truncates first-phase structures in the Itremo Group, and indicates that the fold-and-thrust belt formed prior to ≈565 Ma. Th–U–Pb electron microprobe dating was applied to elongated monazites that lie within the first-phase foliation of Itremo Group metapelites. The detrital cores of zoned monazites reveal two distinct age populations at ∼2000 and 1700 Ma, the latter age giving a maximum depositional age for the Itremo Group. Statistical analysis of ages determined from the rims of zoned monazites and from unzoned monazites indicates three Late Proterozoic–Early Paleozoic monazite growth events at about 565–540, 500 and 430 Ma. The oldest age population is contemporaneous within error, with the intrusion of the dated granite. The two younger age populations are found both in the Th–U–Pb and Ar–Ar data; together with the perturbation of the Rb–Sr system we interpret both ages as due to alteration related to fluid circulation events, possibly connected to the emplacement of pegmatite fields in Central Madagascar. Syn-D1 tectonic growth of contact metamorphism minerals such as andalusite has been observed locally in metapelites along the margin of Middle Neoproterozoic (≈800 Ma) granites, suggesting that D1 in the Itremo Group is contemporaneous with the intrusion of granites at ≈800 Ma. The N-S trending D2 folds are associated with ≈E-W shortening during the final assembly of Gondwana in Late Neoproterozoic–Early Cambrian times.