Uniqueness and examples of compact toric Sasaki-Einstein metrics

In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on $S^5 \sharp k(S^2 \times S^3)$ for each positive integer $k$.Comment: Statements of the results are modifie

Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability

This paper concerns the explicit construction of extremal Kaehler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory. We obtain a characterization, on a large family of projective bundles, of those `admissible' Kaehler classes (i.e., the ones compatible with the bundle structure in a way we make precise) which contain an extremal Kaehler metric. In many cases, such as on geometrically ruled surfaces, every Kaehler class is admissible. In particular, our results complete the classification of extremal Kaehler metrics on geometrically ruled surfaces, answering several long-standing questions. We also find that our characterization agrees with a notion of K-stability for admissible Kaehler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.Comment: 40 pages, sequel to math.DG/0401320 and math.DG/0202280, but largely self-contained; partially replaces and extends math.DG/050151

Tracing the provenance of inherited zircons from peraluminous granites in the Lhasa Terrane and its paleogeographic implications

Peraluminous granites with abundant zircon inheritance are derived from partial melting of Al-rich rocks (e.g. metapelite). Thus the U-Pb age data of inherited zircons from peraluminous granites provide insights into provenance of clastic sediments in their source region, as do the detrital zircons from sedimentary rocks (and their metamorphosed equivalents). This paper reports the whole-rock geochemical and zircon U-Pb geochronological data (95 analyses) of the Early Jurassic peraluminous granites in the central Lhasa subterrane. These data, in combination with the existing data of inherited zircons (104 analyses) from the Permian and Late Triassic peraluminous granites currently available in the central Lhasa subterrane, are used to characterize the inherited zircon signature of the Lhasa Terrane. These granites belong to strongly peraluminous S-type granites, which contain abundant inherited zircons that define two main age populations of 1250~1100Ma (peak at 1181±14Ma) and 550~450Ma (peak at 494±7Ma), comparable to the ca. 1170Ma age population defined by detrital zircons from Paleozoic sedimentary rocks and the emplacement timing of Cambrian volcanic rocks in the Lhasa Terrane, respectively. The ca. 1170Ma age population defined by inherited and detrital zircons in the Lhasa Terrane differs significantly from the age distributions (peak at ca. 960Ma) defined by detrital zircons from Neoproterozoic-Paleozoic sedimentary rocks in the western Qiangtang, Amdo, and Tethyan Hiamalaya in southern Tibet. We propose that the ca. 1181Ma inherited zircons from peraluminous granites in the central Lhasa subterrane were most likely derived from the Albany-Fraser orogenic belt in southwestern Australia and Wilkes Province in East Antarctica, as do the coeval detrital zircons from Paleozoic sedimentary rocks in the Lhasa Terrane, and that the ca. 494Ma inherited zircons might have been sourced from both the Western Australia and Lhasa Terrane itself. This paper provides evidence of U-Pb dating on inherited zircons from peraluminous granites for the paleogeographic connection between the Lhasa Terrane and northern Australia. Our studies on the geology of the Lhasa Terrane indicate that a combined in-situ U-Pb dating on inherited zircons from peraluminous granites and detrital zircons from Paleozoic sedimentary rocks can provide important constraints on paleogeography and tectonomagmatic evolution of other microcontinents along the northern margin of Gondwana

Structure and performance of anisotropic nanocrystalline Nd-Fe-B magnets fabricated by high-velocity compaction followed by deformation

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