Holography and Stiff-matter on the Brane

Recently, Verlinde noted a surprising similarity between Friedmann equation governing radiation dominated universe and Cardy's entropy formula in conformal field theory. In this note, we study a brane-universe filled with radiation and stiff-matter. We analyze Friedmann equation in this context and compare our results with Cardy's entropy formula.Comment: 6 pages, Latex, references added to match with published versio

On the geometry of the symmetrized bidisc

We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $\mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain $$\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}$$ in Isaev's list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $\mathcal D_1$. Among the results on $\mathcal D_1$, of particular interest is the fact that $\mathcal D_1$ is a "symmetrization". When we symmetrize (appropriately defined in the context in the last section) either $\Omega_1$ or $\mathcal{D}^{(2)}_1$ (Isaev's notation), we get $\mathcal D_1$. These two domains $\Omega_1$ and $\mathcal{D}^{(2)}_1$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb{D} \times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $\mathbb{D} \times \mathbb{D}$.Comment: 22 pages, Accepted in Indiana University Mathematics Journa

Pick interpolation and invariant distances

In this article, we study the role of invariant distances in the Pick interpolation problem. Given a Carath\'eodory hyperbolic domain $\Omega$ in some $\mathbb{C}^m$, we have introduced a notion of an invariant object that gives a necessary and sufficient condition for any Pick interpolation problem to be solvable on $\Omega$. This invariant object plays the same role as that of Carath\'eodory pseudodistance in the two-point Pick interpolation problem. Furthermore, a full description of the invariant object is given when $\Omega$ is the open unit disc.Comment: 10 pages, comments are welcom

Black Holes in 4D4D AdS Einstein Gauss Bonnet Gravity With Power- Yang Mills Field

In this paper we construct an exact spherically symmetric black hole solution with a power Yang-Mills (YM) source in the context of $4D$ Einstein Gauss-Bonnet gravity ($4D$ EGB). We choose our source as $(F_{\mu\nu}^{(a)}F^{\mu\nu(a)})^q$, where $q$ is an arbitrary positive real number. Thereafter we study the horizon structure, thermodynamic issues like thermal stability and black hole phase transition of this black hole solution. Our focus here is to analyse the black hole space-time under the net non-linear effect coming both from the gravitational sector (due to Gauss-Bonnet term) as well as from the gauge fields (the power of Yang-Mills field invariant) in $4$-dimensions. We evaluate some extended thermodynamic quantities such as pressure, temperature, entropy in order to establish the form of the Smarr formula and the first law of thermodynamics. The behaviour of heat capacity as a function of horizon radius is thoroughly studied to understand the thermal stability of the black hole solution. An interesting phenomena of existence/ absence of thermal phase transition occur due to the nonlinearity of YM source. For some values of the parameters, we find that the solution exhibits a first-order phase transition, like a van der Waals fluid. In addition, we also verify Maxwell's equal area law numerically by crucial analysis of Gibbs free energy as a function of temperature. Moreover, the critical exponents are derived and showed the universality class of the scaling behaviour of thermodynamic quantities near criticality.Comment: 20 Pages, 16 figures, typos corrected, references added, minor change