Related papers: Random conformal welding for finitely connected re…

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of $\beta X$ where $X$ is the restriction of the…

We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the…

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…

In this article, we use the framework of "Random conformal weldings" (by K. Astala, P. Jones, A. Kupiainen, E. Saksman) to prove the existence of Lehto-welding for the inverse for $\gamma<0.1818$ and independent copies for…

We construct a (non-removable) Jordan curve $\Gamma$ and a non-M\"{o}bius homeomorphism of the Riemann sphere which is conformal on the complement of $\Gamma$ and maps the curve $\Gamma$ onto itself. The curve is flexible in the sense of…

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of…

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $\Omega$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $\Omega$…

The seminal work of Sheffield showed that when random surfaces called Liouville quantum gravity (LQG) are conformally welded, the resulting interface is Schramm-Loewner evolution (SLE). This has been proved for a variety of configurations,…

We give a simple set of geometric conditions on curves $\eta$, $\tilde{\eta}$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $\varphi \colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $\eta$ with $\varphi(\eta)…

Let $M$ be a Riemann surface which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $\Omega \subset \mathbb{C}$. We describe $M$ in terms of $\Omega$ under various assumptions on the boundary…

We show that a conformal connection on a closed oriented surface $\Sigma$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it…

We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in…

Let $U$ be a multiply connected domain of the Riemann sphere $\hat{C}$ whose complement $\hat{C}\setminus U$ has $N<\infty$ components. We show that every conformal map on $U$ can be written as a composition of $N$ maps conformal on simply…

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of…

In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $\Omega$ and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$…

We study the conformal type of surfaces spread over the sphere via random quasiconformal maps. Constructing a random Beltrami coefficient on the complex plane, we obtain a locally quasiconformal homeomorphism with prescribed dilatation that…

We find an extremal problem for conformal maps on a finitely connected subregion of the Riemann sphere containing the point at infinity whose unique solution is a map onto a square domain, that is, a domain whose complementary components…

Let $N=(\Omega,\sigma)$ and $M=(\Omega^*,\rho)$ be doubly connected Riemann surfaces and assume that $\rho$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms…

A large class of Jordan curves on the Riemann sphere can be encoded by circle homeomorphisms via conformal welding, among which we consider the welding homeomorphism of the random SLE loops and the Weil-Petersson class of quasicircles. It…

We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2…