Doubling measures, monotonicity, and quasiconformality

We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.Comment: 20 pages. Revised to address referee's comment

Extremal black holes in D=4 Gauss-Bonnet gravity

We show that four-dimensional Einstein-Maxwell-Dilaton-Gauss-Bonnet gravity admits asymptotically flat black hole solutions with a degenerate event horizon of the Reissner-Nordstr\"om type $AdS_2\times S^2$. Such black holes exist for the dilaton coupling constant within the interval $0\leq a^2<a^2_{\rm cr}$. Black holes must be endowed with an electric charge and (possibly) with magnetic charge (dyons) but they can not be purely magnetic. Purely electric solutions are constructed numerically and the critical dilaton coupling is determined $a_{\rm cr}\simeq 0.488219703$. For each value of the dilaton coupling $a$ within this interval and for a fixed value of the Gauss--Bonnet coupling $\alpha$ we have a family of black holes parameterized by their electric charge. Relation between the mass, the electric charge and the dilaton charge at both ends of the allowed interval of $a$ is reminiscent of the BPS condition for dilaton black holes in the Einstein-Maxwell-Dilaton theory. The entropy of the DGB extremal black holes is twice the Bekenstein-Hawking entropy.Comment: New material and references added, errors corrected including higher decimals in a_cr, figures improve

Vacuum Interpretation for Supergravity M-Branes

A non-local classical duality between the three-block truncated 11D supergravity and the 8D vacuum gravity with two commuting Killing symmetries is established. The supergravity four-form field is generated via an inverse dualisation of the corresponding Killing two-forms in six dimensions. 11D supersymmetry condition is shown to be equivalent to existence of covariantly constant spinors in eight dimensions. Thus any solution to the vacuum Einstein equations in eight dimensions depending on six coordinates and admitting Killing spinors have supersymmetric 11D-supergravity counterparts. Using this duality we derive some new brane solutions to 11D-supergravity including 1/4 supersymmetric intersecting M-branes with a NUT parameter and a dyon solution joining the M2 and M5-branes intersecting at a point.Comment: 4 pages, latex, no figure

Bertotti-Robinson solutions of D=5 Einstein-Maxwell-Chern-Simons-Lambda theory

We present a series of new solutions in five-dimensional Einstein-Maxwell-Chern-Simons theory with an arbitrary Chern-Simons coupling $\gamma$ and a cosmological constant $\Lambda$. For general $\gamma$ and $\Lambda$ we give various generalizations of the Bertotti-Robinson solutions supported by electric and magnetic fluxes, some of which presumably describe the near-horizon regions of black strings or black rings. Among them there is a solution which could apply to the horizon of a topological AdS black ring in gauged minimal supergravity. Others are horizonless and geodesically complete. We also construct extremal asymptotically flat multi-string solutions for $\Lambda=0$ and arbitrary $\gamma$.Comment: 17 pages, revtex