Two derivative Jackiw Teitelboim gravity theory captures the near horizon dynamics of higher dimensional extremal black holes, which is governed by a Schwarzian action at boundary in region. The partition function corresponding to this correctly gives statistical entropy hole. In paper, we study thermodynamics spherically symmetric four holes presence arbitrary perturbative corrections. We find...

If \(\Omega\) is a simply connected domain in \(\overline{\mathbf C}\) then, according to the Ahlfors-Gehring theorem, quasidisk if and only there exists sufficient condition for univalence of holomorphic functions relation growth their Schwarzian derivative. We extend this theorem harmonic mappings by proving criterion on quasidisks. also show that satisfying admit homeomorphic extension and, ...

We prove that if V n is a Chebyshev system on the circle and f(x) is a continuous function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S that takes f to a function L-orthogonal to V . We also prove that if f(x) is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of...

TO THE RIEMANN MAP Zheng-Xu He and Oded Schramm Abstract. Let $ C be a simply connected domain. The Rodin-SullivanTheorem states that a sequence of disk packings in the unit disk U converges, in a well de ned sense, to a conformal map from to U . Moreover, it is known that the rst and second derivatives converge as well. Here, it is proven that for hexagonal disk packings the convergence is C1 ...

Gehring and Pommerenke have shown that if the Schwarzian derivative S f of an analytic function f in the unit disk D satisfies ISf(z)] ~_ 2(1 I z [2 ) -2, then f (D) is a Jordan domain except when f (D) is the image under a M6bius transformation of an infinite parallel strip. The condition ISf(z)l <_ 2(1 lzl2) -2 is the classical sufficient condition for univalence of Nehari. In this paper we s...

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to a...