Related papers: Lappan's five-point theorem for {\phi}-Normal Harm…

We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove…

Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2\pi)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain…

We prove that wave maps that factor as $\mathbb{R}^{1+d} \overset{\varphi_{\text{S}}}{\to} \mathbb{R} \overset{\varphi_{\text{I}}}{\to} M$, subject to a sign condition, are globally nonlinear stable under small compactly supported…

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with…

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion.…

We investigate the univalency and the directional convexity of the convolution $\phi\tilde{*}f=\phi*h+\overline{\phi*g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $\phi$ whose convolution with the mapping…

Assume that $p\in(1,\infty]$ and $u=P_{h}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq…

Let $\phi$ be analytic and univalent ({\it i.e.,} one-to-one) in $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $\phi(\mathbb{D})$ has positive real part, is symmetric with respect to the real axis, starlike with respect to $\phi(0)=1,$…

Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable} if…

In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal F$ in a domain $D\subset \mathbb C,$ and for a…

Suppose $\alpha>-1$ and $1\leq p \leq \infty$. Let $f=P_{\alpha}[F]$ be an $\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2\pi)$, where…

We describe work on solutions of certain non-divergence type and therefore non-variational elliptic and parabolic systems on manifolds. These systems include Hermitian and affine harmonics which should become useful tools for studying…

We consider polyharmonic maps $\phi:(M,g)\rightarrow $\mathbb{E}^n$ of order k from a complete Riemannian manifold into the Euclidean space and let $p$ be a real constant satisfying $1<p<\infty$. (i) If, $\int_M|W^{k-1}|^p dv_g<\infty,$ and…

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $\psi:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $\phi:\s^3\to \s^3$. We…

We show that on any Riemannian manifold with H\"older continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry.…

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the…

Suppose that $f$ satisfies the following: $(1)$ the polyharmonic equation $\Delta^{m}f=\Delta(\Delta^{m-1} f)$$=\varphi_{m}$ $(\varphi_{m}\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions…

\noindent In the present investigation, we find the sharp bound of fifth coefficient of analytic normalized function $f$ satisfying $z f'(z)/f(z) \prec \varphi(z)$ when coefficients of $\varphi$ satisfy certain conditions. For an…

We construct explicit examples of Dirac-harmonic maps $(\phi, \psi)$ between Riemannian manifolds $(M,g)$ and $(N,g')$ which are non-trivial in the sense that $\phi$ is not harmonic. When $\dim M=2$, we also produce examples where $\phi$ is…

The first result in this study is a non-existence theorem for $\alpha-$harmonic mappings. Additionally, a direct connection between the $\alpha-$ harmonic and harmonic maps is made possible via conformal deformation. Second, the instability…