We provide a simple coordinatization for the group G2, which is analogous to the Euler coordinatization for SU (2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H. * cacciatori@mi.infn.it † BLCerchiai@lbl.gov ‡ alberto.dellavedova@unimib.it § giovanni.ortenzi@unimib.it ¶ Graduate visitor (visitatore laureato), ascotti@mindspring.com

During the 1999^2000 Italian Expedition, an airborne radar survey was performed along12 transects across LakeVostok, Antarctica, and its western and eastern margins. Ice thickness, subglacial elevation and the precise location of lake boundaries were determined. Radar data confirm the geometry derived from previous surveys, but with some slight differences. We measured a length of up to 260 km, a maximum width of 81km and an area of roughly 14 000 km 2 . Along the major axis, from north to south, the ice thickness varies from 3800 to 4250 m, with a decreasing gradient. From west to east the ice thickness is fairly constant, except for two narrow strips located on the western and eastern margins, where it increases with high thickening rate. Over the lake the surface elevation increases from 3476 m a.s.l. (south) to 3525 (north), with a decreasing gradient, while the lake surface elevation decreases from^315 to^750 m a.s.l., with a decreasing gradient (absolute value). The icesurface and lake-ceiling slopes suggest that the lake is in a state of hydrostatic equilibrium.

We construct in an explict way the soliton equation corresponding to the affine Kac-Moody Lie algebra G (1) 2 together with their bihamiltonian structure. Moreover the Riccati equation satisfied by the generating function of the commuting Hamiltonians densities is also deduced. Finally we describe a way to deduce the bihamiltonian equations directly in terms of this latter functions

Abstract. In this paper we extend the notion of Futaki invariant to big and nef classes so as to define a continuous function on the Kähler cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities are sufficient to check K-semistability. A similar improvement on Donaldson's lower bound for Calabi energy is given.