Related papers: Schwarz lemma for harmonic mappings between Rieman…

We consider the space of geodesic laminations on a surface, endowed with the Hausdorff metric d_H and with a variation of this metric called the d_log metric. We compute and/or estimate the Hausdorff dimensions of these two metrics. We also…

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first…

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann…

We consider the leapfrog algorithm by Noakes for computing geodesics on Riemannian manifolds. The main idea behind this algorithm is to subdivide the original endpoint geodesic problem into several local problems, for which the endpoint…

In this note we prove a version of the classical Schwarz lemma for the first eigenvalues of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne…

Motivated by recent papers \cite{For-Rong 2021} and \cite{Ng-Rong 2024} we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs…

We establish that the intersection number between the horizontal foliations of any two finite-area holomorphic quadratic differentials on an arbitrary Riemann surface is finite. Our main result shows that the intersection number is jointly…

We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total…

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz…

We construct equivariant harmonic maps between cohomogeneity one manifolds.

We give a proof of the classical Schwarz reflection principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(\kappa, \tau)$ for $\kappa \leqslant 0$ and $\tau \geqslant 0$. In our previous paper we proved a…

We prove an intersection formula for two plane branches in terms of their semigroups and key polynomials. Then we provide a strong version of Bayer's theorem on the set of intersection numbers of two branches and apply it to the logarithmic…

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous…

In this note we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that modulo M\"obius transformations the moduli space of such local integrals (if nonempty) is either…

We investigate horizontal conformality of a differential of a map between Riemannian manifolds where the tangent bundles are equipped with Cheeger--Gromoll type metrics. As a corollary, we characterize the differential of a map as a…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…

In this paper, we presents a method for factoring morphisms between arithmetic surfaces based on the regularity of arithmetic surfaces. Using this factorization, we derive a Riemann-Hurwitz formula satisfied by the ramification divisor and…

The aim of this article is to give an elementary proof of the fact that the Schwarz-Pick Lemma follows from the Ahlfors-Schwarz-Pick Lemma.

With an easy application of maximum principle, we establish a Schwarz-type lemma for squeezing function on finitely connected planar domains that directly yields the explicit formula for squeezing function on doubly connected domains…

This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.