The Effect of Symmetry on the Riemann Map

The Riemann mapping theorem guarantees the existence of a conformal mapping or Riemann map in the complex plane from the open unit disk onto an open simply-connected domain, which is not all of the complex plane. Although its existence is guaranteed, the Riemann map is rarely known except for special domains like half-planes, strips, etc. Therefore, any information we can determine about the Riemann map for any class of domains is interesting and useful. This research investigates how symmetry affects the Riemann map. In particular, we define domains with symmetries called Rectangular Domains or RDs. The Riemann map of an RD has real-valued coefficients, as opposed to complex-valued, and therefore we can determine the sign of the coefficients of the Taylor series about the origin of the Riemann map, f(z), from the unit disk onto RDs determined by f(0)=0 and f \u27(0)\u3e0. We focus on the form of the Riemann map for specific RD polygons. These include rhombi, rectangles and non-equilateral octagons which have 2-fold symmetries. We also investigate equilateral polygons with more than 2-fold symmetries such as squares (rotated diamond), regular polygons, and equilateral octagons

Islands with Gravitating Baths: Towards ER = EPR

We study the Page curve and the island rule for black holes evaporating into gravitating baths, with an eye towards establishing a connection with the ER=EPR proposal. We consider several models of two entangled 2d black holes in Jackiw-Teitelboim (JT) gravity with negative cosmological constant. The first, "doubled PSSY," model is one in which the black holes have end-of-the-world (ETW) branes with a flavour degree of freedom. We study highly entangled states of this flavour degree of freedom and find an entanglement-induced Hawking-Page-like transition from a geometry with two disconnected black holes to one with a pair of black holes connected by a wormhole, thus realising the ER = EPR proposal. The second model is a dynamical one in which the ETW branes do not have internal degrees of freedom but the JT gravity is coupled to a 2d CFT, and we entangle the black holes by coupling the two CFTs at the $AdS$ boundary and evolving for a long time. We study the entanglement entropy between the two black holes and find that the story is substantially similar to that with a non-gravitating thermal bath. In the third model, we couple the two ends of a two-sided eternal black hole and evolve for a long time. Finally, we discuss the possibility of a Hawking-Page-like transition induced by real-time evolution that realises the ER = EPR proposal in this dynamical setting.Comment: 47+14 pages, 16 figures. Comments welcom