Schwarz lemma for harmonic mappings into a geodesic line in a Riemann surfaces

2020 ◽  
Vol 66 (2) ◽  
pp. 275-282
Author(s):  
David Kalaj
Keyword(s):  
Riemann Surfaces ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Geodesic Line

scholarly journals A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Jiaolong Chen ◽  
David Kalaj
Keyword(s):  
Unit Ball ◽  
Explicit Form ◽  
Smooth Function ◽  
Harmonic Mapping ◽  
Sharp Inequality ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Manuscripta Math ◽  
Sharp Constant ◽  
Mapping Theory

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

scholarly journals Some Properties and Regions of Variability of Affine Harmonic Mappings and Affine Biharmonic Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Analytic Functions ◽  
Schwarz Lemma ◽  
Harmonic Mappings

We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions (their boundaries) are determined explicitly and the proofs rely on Schwarz lemma or subordination.

scholarly journals The value distribution of harmonic mappings between Riemannian n-spaces

1975 ◽  
Vol 59 ◽  
pp. 45-58
Author(s):  
Hideo Imai
Keyword(s):  
Riemann Surface ◽  
Kernel Function ◽  
Riemann Surfaces ◽  
Distribution Theory ◽  
Value Distribution ◽  
Harmonic Mappings ◽  
Value Distribution Theory ◽  
Theoretic Method ◽  
Analytic Mapping ◽  
Crucial Part

We are concerned with the value distribution of a mapping of an open Riemannian n-space (n ≧ 3) into a Riemannian n-space. The value distribution theory of an analytic mapping of Riemann surfaces was initiated by S. S. Chern [1] and developed mainly by L. Sario [8], [9], [10], [11], and then by H. Wu [14], [15]. The most crucial part in Sario’s theory is the introduction of a kernel function on an arbitrary Riemann surface to describe appropriately the proximity of two points. His method indicates that the potential theoretic method is one of the powerful methods in the value distribution theory.

scholarly journals Harmonic mappings and moduli spaces of Riemann surfaces

2009 ◽  
Vol 14 (1) ◽  
pp. 171-196 ◽  
Author(s):  
Jürgen Jost ◽  
Shing Tung Yau
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Harmonic Mappings

scholarly journals Quasihyperbolic quasi-isometry and Schwarz lemma of planar flat harmonic mappings

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5371-5383
Author(s):  
Qingtian Shi ◽  
Yi Qi
Keyword(s):  
Quasiconformal Mapping ◽  
Harmonic Mapping ◽  
Schwarz Lemma ◽  
Harmonic Mappings ◽  
Sufficient Condition

A sufficient condition of a flat harmonic quasiconformal mapping to be a quasihyperbolic quasiisometry on any subdomain of C is given in this paper, which generalizes the corresponding results of Euclidean and 1/|?|2 harmonic mappings. As an application, Schwarz lemma of flat harmonic mapping is also investigated. Besides, properties and constructions of flat harmonic mapping are obtained at the same time.

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