From 2-Dimensional Surfaces to Cosmological Solutions

We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.Comment: 12 pages, no figures, LaTeX2e, to be published in Class. Quant. Gra

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Backlund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalisation of the integrable discrete Tzitzeica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretisation of affine spheres in affine differential geometry

D-dimensional metrics with D-3 symmetries

Hidden symmetry transformations of D-dimensional vacuum metrics with D-3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We relate the 4-dimensional Reissner-Nordstr\"om solution to a particular case of the 5-dimensional Gross-Perry metric.Comment: 8 page

Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates

Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebański's first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4 + 4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries

Near horizon geometries and black hole holograph

Two quasi local approaches to black holes are combined: Near Horizon Geometries (NHG) and stationary Black Hole Holographs (BHH). Necessary and sufficient conditions on BHH data for the emergence of NHGs as resulting vacuum solutions to Einstein's equations are found

GPR54 regulates ERK1/2 activity and hypothalamic gene expression in a Galpha(q/11) and beta-arrestin-dependent manner

G protein-coupled receptor 54 (GPR54) is a G(q/11)-coupled 7 transmembrane-spanning receptor (7TMR). Activation of GPR54 by kisspeptin (Kp) stimulates PIP(2) hydrolysis, Ca(2+) mobilization and ERK1/2 MAPK phosphorylation. Kp and GPR54 are established regulators of the hypothalamic-pituitary-gonadal (HPG) axis and loss-of-function mutations in GPR54 are associated with an absence of puberty and hypogonadotropic hypogonadism, thus defining an important role of the Kp/GPR54 signaling system in reproductive function. Given the tremendous physiological and clinical importance of the Kp/GPR54 signaling system, we explored the contributions of the GPR54-coupled G(q/11) and beta-arrestin pathways on the activation of a major downstream signaling molecule, ERK, using G(q/11) and beta-arrestin knockout mouse embryonic fibroblasts. Our study revealed that GPR54 employs the G(q/11) and beta-arrestin-2 pathways in a co-dependent and temporally overlapping manner to positively regulate ERK activity and pERK nuclear localization. We also show that while beta-arrestin-2 potentiates GPR54 signaling to ERK, beta-arrestin-1 inhibits it. Our data also revealed that diminished beta-arrestin-1 and -2 expression in the GT1-7 GnRH hypothalamic neuronal cell line triggered distinct patterns of gene expression following Kp-10 treatment. Thus, beta-arrestin-1 and -2 also regulate distinct downstream responses in gene expression. Finally, we showed that GPR54, when uncoupled from the G(q/11) pathway, as is the case for several naturally occurring GPR54 mutants associated with hypogonadotropic hypogonadism, continues to regulate gene expression in a G protein-independent manner. These new and exciting findings add significantly to our mechanistic understanding of how this important receptor signals intracellularly in response to kisspeptin stimulation