Related papers: Factorisation of conformal maps on finitely connec…
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…
The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map…
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…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is…
We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function.…
We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain $D \subset \hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connected subdomains $D_j$…
We study conformal maps from multiply connected domains in the extended complex plane onto lemniscatic domains. Walsh proved the existence of such maps in 1956 and thus obtained a direct generalization of the Riemann mapping theorem to…
Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…
In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by…
In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a…
Let X be a closed surface of genus two embedded in the 3-sphere. Then X inherits a metric and an orientation, which give an almost complex structure, which automatically integrates to a genuine complex structure, making X a Riemann surface.…
We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.
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 present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations…
A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a…
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian…
In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization…
It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes…
Let M be a compact Riemannian manifold and let $\mu$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian…