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.

The Interplay between Melting, Refertilization and Carbonatite Metasomatism in Off-Cratonic Lithospheric Mantle under Zealandia: an Integrated Major, Trace and Platinum Group Element Study

It is widely accepted that stabilization of the continental crust requires the presence of sub-continental lithospheric mantle. However, the degree of melt depletion required to stabilize the lithosphere and whether widespread refertilization is a significant process remain unresolved. Here, major and trace element, including platinum group elements (PGE), characterization of 40 mantle xenoliths from 13 localities is used to constrain the melt depletion, refertilization and metasomatic history of lithospheric mantle underneath the micro-continent Zealandia. Our previously published Re–Os isotopic data for a subset of these xenoliths indicate Phanerozoic to Paleoproterozoic ages and, reinterpreted with the new major and trace element data presented here, demonstrate that a large volume (>2 million km3) of lithospheric mantle with an age of 1·99 ± 0·21 Ga is present below the much younger crust of Zealandia. A peritectic melting model using moderately incompatible trace elements (e.g. Yb) in bulk-rocks demonstrates that these peridotites experienced a significant range of degrees of partial melting, between 3 and 28%. During subsolidus equilibration clinopyroxene gains significant rare earth elements (REE), which then leads to the underestimation of the degree of partial melting by ≤12% in fertile xenoliths. A new approach taking into account the effects of subsolidus re-equilibration on clinopyroxene composition effectively removes discrepancies in the calculated degree of melting and provides consistent estimates of between 4 and 29%. The estimated amount of melting is independent of the Re–Os model ages of the samples. The PGE patterns record simple melt depletion histories and the retention of primary base metal sulfides in the majority of the xenoliths. A rapid decrease in Pt/IrN observed at c. 1·0 wt % Al2O3 is a direct result of the exhaustion of sulfide in the mantle residue at c. 20–25% partial melting and the inability of Pt to form a stable alloy phase. Major elements preserve evidence for refertilization by a basaltic component that resulted in the formation of secondary clinopyroxene and low-forsterite olivine. The majority of xenoliths show the effects of cryptic metasomatic overprinting, ranging from minor to strong light REE enrichments in bulk-rocks (La/YbN = 0·16–15·9). Metasomatism is heterogeneous, with samples varying from those with weak REE enrichment and notable positive Sr and U–Th anomalies and negative Nb–Ta anomalies in clinopyroxene to those that have extremely high concentrations of REE, Th–U and Nb. Chemical compositions are consistent with a carbonatitic component contributing to the metasomatism of the lithosphere under Zealandia. Notably, the intense metasomatism of the samples did not affect the PGE budget of the peridotites as this was controlled by residual sulfides.