1 The Dirichlet problem of harmonic mappings

For the coordinates \((x^1,x^2)\) we define the domain

$$\begin{aligned} \Omega _1:=\{X=(x^1,x^2)\in \mathbb R^2\,:\, |X|<1\} \end{aligned}$$

and introduce the unit disc

$$\begin{aligned} B:=\{w=u+iv\in \mathbb C\,:\ |w|<1\} \end{aligned}$$

with the parameters \(u+iv\cong (u,v)\). Now we prescribe the Riemannian metric

$$\begin{aligned} \displaystyle ds^2= & {} \sum _{j,k=1,2} g_{jk}(x^1,x^2)\,dx^j\,dx^k\nonumber \\ \displaystyle= & {} g_{11}(x^1,x^2)\,(dx^1)^2+2g_{12}(x^1,x^2)\,dx^1\,dx^2 +g_{22}(x^1,x^2)\,(dx^2)^2 \end{aligned}$$
(1.1)

on the disc \(\Omega _1\). Here we require our coefficients to satisfy

$$\begin{aligned}&g_{jk}=g_{jk}(x^1,x^2)\in C^{1+\alpha }(\overline{\Omega }_1,\mathbb R)\quad \text{ for }\quad j,k=1,2,\nonumber \\&\quad g_{12}(x^1,x^2)=g_{21}(x^1,x^2)\quad \text{ in }\quad \overline{\Omega }_1, \end{aligned}$$
(1.2)
$$\begin{aligned}&\displaystyle \lambda \,|\xi |^2\,\le \sum _{j,k=1,2} g_{jk}(x^1,x^2)\xi ^j\xi ^k\le \frac{1}{\lambda }\,|\xi |^2\nonumber \\&\quad \text{ for } \text{ all }\quad \xi =(\xi ^1,\xi ^2)\in \mathbb R^2\quad \text{ and }\quad (x^1,x^2)\in \overline{\Omega }_1, \end{aligned}$$
(1.3)

with the Hölder constant \(\alpha \in (0,1)\) and the quantity \(\lambda \in (0,1]\).

By a continuity method the following profound result is established:

Theorem 1

(Conformal mappings w. r. t. Riemannian metrics) For the Riemannian metric (1.1), (1.2), (1.3) there exists a \(C^{2+\alpha }({\overline{B}},\overline{\Omega }_1)\)-diffeomorphic, positive-oriented mapping

$$\begin{aligned} X=X(u,v)=(x^1(u,v),x^2(u,v)):{\overline{B}}\rightarrow \overline{\Omega }_1\in C^{2+\alpha }({\overline{B}},\overline{\Omega }_1) \end{aligned}$$

satisfying the weighted conformality relations

$$\begin{aligned}&\displaystyle \sum _{j,k=1,2} x_u^j(u,v)g_{jk}(x^1(u,v),x^2(u,v))x_v^k(u,v)= 0\nonumber \\&\displaystyle \sum _{j,k=1,2}x_u^j(u,v)g_{jk}(x^1,x^2)x_u^k(u,v)= \sum _{j,k=1,2}x_v^j(u,v)g_{jk}(x^1,x^2)x_v^k(u,v)\,\text{ in }\,B.\nonumber \\ \end{aligned}$$
(1.4)

Proof

See our uniformization theorem from [12] Chap. 12 the Theorem 8.2. \(\square \)

Due to Proposition 7.1 of [12] Chap. 12, the function X then satisfies the nonlinear elliptic system

$$\begin{aligned} \Delta x^l+\sum _{j,k=1,2}\Gamma _{jk}^l(x_u^jx_u^k+x_v^jx_v^k)=0\quad \text{ in }\quad B\quad \text{ for }\quad l=1,2. \end{aligned}$$
(1.5)

Here we use the Christoffel symbols

$$\begin{aligned} \Gamma _{jk}^l:=\frac{1}{2} \sum _{i=1,2}g^{li}(g_{ki,x^j}+g_{ij,x^k}-g_{jk,x^i}),\quad j,k,l=1,2 \end{aligned}$$
(1.6)

with the inverse matrix \((g^{jk})_{j,k=1,2}:=(g_{jk})^{-1}_{j,k=1,2}\). Therefore, X represents a one-to-one harmonic mapping of the disc \(\{B,(\delta _{jk})\}\) with the Euclidean metric \((\delta _{jk})_{j,k=1,2}\) onto the disc \(\{\Omega _1,(g_{jk})\}\). On account of well-known regularity results, the associate boundary function

$$\begin{aligned} \Phi (u,v):=X(u,v),\quad (u,v)\in \partial B\quad \text{ with }\quad \Phi :\partial B\rightarrow \partial \Omega _1\in C^{2+\alpha }(\partial B,\partial \Omega _1) \end{aligned}$$
(1.7)

appearing within this approximation and selection procedure, yields a positive-oriented \(C^{2+\alpha }(\partial B,\partial \Omega _1)\)-diffeomorphism between the circumferences \(\partial B\) and \(\partial \Omega _1\). This weighted-conformal mapping is uniquely determined by a three-point-condition on the boundary. Of course, this boundary representation optimally appears for these weighted-conformal mappings and cannot be prescribed!

Remark 1

Starting with an analogous result to Theorem 1 above, Jost [7] has constructed harmonic diffeomorphisms, for arbitrary convex boundary data, by deformation of the boundary values via a topological method. This has been combined with a priori estimates for their Jacobian by E. Heinz. With the aid of the maximum principle by Jäger and Kaul [6], then Jost obtained the diffeomorphic character of harmonic maps by reconstruction.

In Sect. 4 we shall see directly the one-to-one character of our harmonic maps, established in Theorem 2 below, and may dispense of the uniqueness for the associate Dirichlet problem. Here we prescribe the Riemannian metric (1.1) on the whole plane \(\mathbb R^2\), which is Euclidean outside of the disc

$$\begin{aligned} \Omega _M:=\{X=(x^1,x^2)\in \mathbb R^2:|X|<M\} \end{aligned}$$

of a fixed radius \(0<M<+\infty \). More precisely, we assume that our coefficients satisfy the following conditions with the Hölder constant \(\alpha \in (0,1)\) and a positive number \(\lambda \in (0,1]\) as follows:

$$\begin{aligned}&g_{jk}=g_{jk}(x^1,x^2)\in C^{1+\alpha }(\mathbb R^2,\mathbb R)\quad \text{ for }\quad j,k=1,2,\nonumber \\&g_{12}(x^1,x^2)=g_{21}(x^1,x^2)\quad \text{ in }\quad \mathbb R^2,\nonumber \\&g_{jk}(x^1,x^2)=\delta _{jk}\quad \text{ in }\quad \mathbb R^2{\setminus } \Omega _M\quad \text{ for }\quad j,k=1,2, \end{aligned}$$
(1.8)

and

$$\begin{aligned}&\displaystyle \lambda \,|\xi |^2\,\le \sum _{j,k=1,2} g_{jk}(x^1,x^2)\xi ^j\xi ^k\le \frac{1}{\lambda }\,|\xi |^2\nonumber \\&\quad \text{ for } \text{ all }\quad \xi =(\xi ^1,\xi ^2)\in \mathbb R^2\quad \text{ and }\quad (x^1,x^2)\in \mathbb R^2. \end{aligned}$$
(1.9)

Furthermore, we require that the metric \(ds^2\) possesses a moderate deviation in the disc \(\Omega _M\) from the Euclidean metric with the constant \(\displaystyle a\in (0,\frac{1}{2M})\) in the following sense: The associate Christoffel symbols (1.6) satisfy the estimate

$$\begin{aligned}&\sqrt{\Big (\sum _{j,k=1,2}\Gamma _{jk}^1\xi ^j\xi ^k\Big )^2+\Big (\sum _{j,k=1,2}\Gamma _{jk}^2\xi ^j\xi ^k\Big )^2}\le a |\xi |^2\nonumber \\&\quad \text{ for } \text{ all }\quad \xi =(\xi ^1,\xi ^2)\in \mathbb R^2\quad \text{ and }\quad (x^1,x^2)\in \mathbb R^2. \end{aligned}$$
(1.10)

By the Leray–Schauder degree of mapping we can establish the following

Theorem 2

(Dirichlet problem for moderate harmonic mappings) Let the Riemannian metric (1.1), (1.8), (1.9) be given with a moderate deviation (1.10) by the constant \(\displaystyle a\in (0,\frac{1}{2M})\) from the Euclidean metric. For each boundary function

$$\begin{aligned} \Phi \in C^{2+\alpha }(\partial B,\mathbb R^2)\quad \text{ with }\quad |\Phi (u,v)|\le M,\,\forall \,(u,v)\in \partial B \end{aligned}$$

there exists a solution

$$\begin{aligned} \displaystyle X= & {} X(u,v)=(x^1(u,v),x^2(u,v)):{\overline{B}}\rightarrow \mathbb R^2\in C^{2+\alpha }({\overline{B}},\mathbb R^2)\nonumber \\&\displaystyle \text{ with }\quad |X(u,v)|\le M\quad \text{ for } \text{ all }\quad (u,v)\in {\overline{B}} \end{aligned}$$
(1.11)

for the system (1.5), (1.6) of harmonic mappings under the boundary condition

$$\begin{aligned} X(u,v)=\Phi (u,v)\quad \text{ for } \text{ all }\quad (u,v)\in \partial B. \end{aligned}$$
(1.12)

Proof

From Theorem 4.4 of [12] Chap. 12 we deduce the existence of a harmonic mapping (1.11) under the boundary conditions (1.12). \(\square \)

Remark 2

Due to the geometric maximum principle by E. Heinz (see Theorem 1.4 in [12] Chap. 12), the solution X of Theorem 2 is subject to the inequality

$$\begin{aligned} \sup _{(u,v)\in {\overline{B}}}|X(u,v)|\le \sup _{(u,v)\in \partial B}|X(u,v)| . \end{aligned}$$
(1.13)

When the boundary values satisfy \(|\Phi (u,v)|<M,\,\forall (u,v)\in \partial B\), then the estimate

$$\begin{aligned} \sup _{(u,v)\in {\overline{B}}}|X(u,v)|<M \end{aligned}$$
(1.14)

follows, and \(X:{\overline{B}}\rightarrow \Omega _M\) represents an inner solution of the system (1.5), (1.6), briefly an inner harmonic mapping.

2 Geodesically stable Riemannian domains

We begin our considerations with the central

Definition 1

We call the disc \(\Omega _M\) of radius \(0<M<+\infty \) endowed with a Riemannian metric (1.1), (1.8), (1.9) a geodesically stable Riemannian domain or simply a stable Riemannian domain, if each geodesic—in unit velocity—emanating from an arbitrary boundary point \(X_0\in \partial \Omega _M\) into an interior direction \(\xi \in S^1\)—within the disc \(\Omega _M\) -

$$\begin{aligned} \Big \{Y(t)=Y(t;\xi ,X_0)\in \overline{\Omega _M},\quad 0\le t \le \tau (X_0,\xi )\Big \} \end{aligned}$$
(2.1)

of the lenght \(\tau (X_0,\xi )>0\) does not contain conjugate points.

Remark 3

Let the Riemannian metric (1.1), (1.8), (1.9) of the regularity class \(C^3\) be given, such that their Gaussian curvature K satisfies

$$\begin{aligned} K(x^1,x^2)\le \kappa ,\quad \forall \,(x^1,x^2)\in \mathbb R^2 \end{aligned}$$
(2.2)

with the barrier \(\kappa \in [0,+\infty )\). Furthermore, let the diameter of the Riemannian domain be bounded by the constant \(\displaystyle \frac{\pi }{\sqrt{\kappa }}\in (0,+\infty ]\) as follows:

$$\begin{aligned} \tau (X_0,\xi )<\frac{\pi }{\sqrt{\kappa }}\quad \text{ for } \text{ all }\quad X_0\in \partial \Omega _M\quad \text{ and } \text{ every } \text{ interior } \text{ direction }\quad \xi \in S^1. \end{aligned}$$
(2.3)

Then this Riemannian domain is necessarily stable.

In this context, we refer our readers to the comparison theorem of J. C. F. Sturm in Satz 3 of Kapitel VII, § 7 from our treatise Analysis [13].

Of central importance is the subsequent

Lemma 1

(Geodesic central fields) Let the domain \(\Omega _M\) be endowed with a stable Riemannian metric from Definition 1, and a boundary point \(X_0\in \partial \Omega _M\) be chosen arbitrarily. Then the family of geodesics

$$\begin{aligned} \mathbf Y(t,s)=\mathbf Y(t,s;X_0),\quad 0< t\le \tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(2.4)

with their initial position

$$\begin{aligned} \mathbf Y(0+,s;X_0)=X_0,\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(2.5)

and their initial velocity

$$\begin{aligned} \mathbf Y_t(0+,s;X_0)=-\exp (is)\cdot \,|X_0|^{-1}\,X_0 ,\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(2.6)

yields a simple covering of the pointed disc \(\overline{\Omega _M}{\setminus }\{X_0\}\).

Proof

For arbitrary \(\displaystyle -\frac{\pi }{2}<s_1<s_2<+\frac{\pi }{2}\) we consider the geodesic

$$\begin{aligned} \mathbf Y(t,s_1)=\mathbf Y(t,s_1;X_0),\quad 0\le t\le \tau (X_0,s_1), \end{aligned}$$
(2.7)

the circular arc

$$\begin{aligned} \mathbf Z(s;X_0)=\mathbf Y(\tau (X_0,s),s;X_0),\quad s_1\le s\le s_2, \end{aligned}$$
(2.8)

and the geodesic

$$\begin{aligned} \mathbf Y(t,s_2)=\mathbf Y(\tau (X_0,s_2)-t,s_2;X_0),\quad 0\le t\le \tau (X_0,s_2). \end{aligned}$$
(2.9)

Since our geodesics do not contain conjugate points within \(\overline{\Omega _M}\), the arcs (2.7) and (2.8) and (2.9) consecutively constitute a Jordan contour \(\Gamma (s_1,s_2;X_0)\) for parameters \(s_1<s_2\) chosen sufficiently near. They form a Jordan curve \(\Gamma (s_1,s_2;X_0)\) for arbitrary parameters \(\displaystyle -\frac{\pi }{2}<s_1<s_2<+\frac{\pi }{2}\) as well, since the mapping

$$\begin{aligned} \mathbf Z(s;X_0)=\mathbf Y(\tau (X_0,s),s;X_0),\quad -\frac{\pi }{2}<s<-\frac{\pi }{2} \end{aligned}$$
(2.10)

is strictly monotonic. Therefore, the interior of the Jordan curves \(\Gamma (s_1,s_2;X_0)\) exhausts the domain \(\Omega _M\) for \(\displaystyle s_1\rightarrow -\frac{\pi }{2}+\) and \(\displaystyle s_2\rightarrow +\frac{\pi }{2}-\). These contours \(\Gamma (s_1,s_2;X_0)\) cover \(\partial \Omega _M\) in the limit \(\displaystyle s_1= -\frac{\pi }{2}\) and \(\displaystyle s_2= +\frac{\pi }{2}\), where the singularity \(X_0\) remains fixed. \(\square \)

As in [13] Kap. VII, Sect. 5 we introduce

Definition 2

The Riemannian inner product of the planar vector fields \(Y(t)=(y_1(t),y_2(t))\) and \(Z(t)=(z_1(t),z_2(t))\) along the plane curve \(X(t)=(x_1(t),x_2(t))\) with the parameter \(a<t<b\) is determined as follows:

$$\begin{aligned} \Big [Y(t),Z(t)\Big ]_{X(t)}:=\sum _{j,k=1,2}g_{jk}(X(t))y_j(t)z_k(t),\quad a<t<b. \end{aligned}$$
(2.11)

Remark 4

From the Gauß–Riemann-Lemma (see Satz 2 in [13] Kap. VII, Sect. 4), we realize the following identities for our geodesic central field in Lemma 1 above:

$$\begin{aligned}&\displaystyle \Big [\mathbf Y_t(t,s),\mathbf Y_t(t,s)\Big ]_{\mathbf Y(t,s)}=1,\quad G(t,s):=\Big [\mathbf Y_s(t,s),\mathbf Y_s(t,s)\Big ]_{\mathbf Y(t,s)}>0,\nonumber \\&\quad \displaystyle \Big [\mathbf Y_t(t,s),\mathbf Y_s(t,s)\Big ]_{\mathbf Y(t,s)}=0\quad ;\quad 0< t\le \tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}. \end{aligned}$$
(2.12)

Definition 3

Let the stable Riemannian domain \(\Omega _M\) of Definition 1 with the geodesic central fields (2.4) of Lemma 1 be given. For all points \(X\in \Omega _M\) with their unique representation

$$\begin{aligned} X=\mathbf Y(t,s)=\mathbf Y(t,s;X_0),\quad 0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(2.13)

we define the lifted vector fields

$$\begin{aligned}&\displaystyle \widehat{\mathbf Y_t}(x^1,x^2;X_0)=\widehat{\mathbf Y_t}(X;X_0):=\mathbf Y_t(t,s),\quad X=(x^1,x^2)\in \Omega _M,\nonumber \\&\displaystyle \widehat{\mathbf Y_s}(x^1,x^2;X_0)=\widehat{\mathbf Y_s}(X;X_0):=\mathbf Y_s(t,s),\quad X=(x^1,x^2)\in \Omega _M \end{aligned}$$
(2.14)

and the lifted Gaussian fundamental coefficient

$$\begin{aligned} {\widehat{G}}(x^1,x^2;X_0)={\widehat{G}}(X;X_0):=G(t,s),\quad X=(x^1,x^2)\in \Omega _M. \end{aligned}$$
(2.15)

We obtain with

$$\begin{aligned} \Bigg \{ \widehat{\mathbf Y_t}(X;X_0),\,\frac{\widehat{\mathbf Y_s}(X;X_0)}{\sqrt{{\widehat{G}}(X;X_0)}}\Bigg \},\quad X=(x^1,x^2)\in \Omega _M \end{aligned}$$
(2.16)

the Gaussian geodesic frame, which constitutes an orthonormal, positive-oriented system of vectors—with respect to the Riemannian inner product (2.11)—on account of Remark 4 above.

In stable Riemannian domains, we can conveniently characterize the convex hull of arbitrary compact sets \(F\subset \Omega _M\) with the subsequent

Definition 4

Let the stable Riemannian domain \(\Omega _M\) of Definition 1 with the geodesic central fields (2.4) of Lemma 1 be given. For all \(\displaystyle \,-\frac{\pi }{2}<s_0<+\frac{\pi }{2}\,\) we introduce the geodesic region

$$\begin{aligned} \Theta (s_0;X_0):=\Big \{\mathbf Y(t,s;X_0)\in \Omega _M\Big |\,0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s\le s_0\Big \}. \end{aligned}$$
(2.17)

This region is closed by the geodesic arc

$$\begin{aligned} \mathbf Y(\tau (X_0,s_0)-t,\,s_0)=\mathbf Y(\tau (X_0,s_0)-t,s_0;X_0),\quad 0\le t\le \tau (X_0,s_0) \end{aligned}$$
(2.18)

and furthermore bounded by the circular arc

$$\begin{aligned} \mathbf Z(s;X_0)=\mathbf Y(\tau (X_0,s),s;X_0),\quad -\frac{\pi }{2}< s < s_0, \end{aligned}$$
(2.19)

where these curves constitute with \(\partial \Theta (s_0;X_0)\) a positive-oriented Jordan contour. For an arbitrary compact set \(F\subset \Omega _M\), we define the convex hull \({{\mathcal {H}}}(F)\) of F within the stable Riemannian domain \(\Omega _M\) as follows:

$$\begin{aligned} {{\mathcal {H}}}(F):=\bigcap \Big \{\Theta (s_0;X_0)\Big |\,X_0\in \partial \Omega _M,\,s_0\in (-\frac{\pi }{2},+\frac{\pi }{2}):\,F\subset \Theta (s_0;X_0)\Big \}. \end{aligned}$$
(2.20)

Finally, we introduce the important geodesic function with

Definition 5

Let the stable Riemannian domain \(\Omega _M\) of Definition 1 with the geodesic central fields (2.4) of Lemma 1 be given. For all points \(X\in \overline{\Omega _M}{\setminus } \{X_0\}\) with their unique representation

$$\begin{aligned} X=\mathbf Y(t,s)=\mathbf Y(t,s;X_0),\quad 0< t\le \tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(2.21)

we define the geodesic function

$$\begin{aligned} \Psi (x^1,x^2;X_0)=\Psi (X;X_0):=s\in (-\frac{\pi }{2},+\frac{\pi }{2}),\quad X=(x^1,x^2)\in \overline{\Omega _M}{\setminus } \{X_0\}. \end{aligned}$$
(2.22)

Remark 5

Obviously, the equation \(\,\Psi (X;X_0)=s_0,\,X\in \Omega _M\,\) describes the geodesic

$$\begin{aligned} \mathbf Y(t,s_0;X_0),\quad 0< t< \tau (X_0,s_0) \end{aligned}$$

for all \(\displaystyle -\frac{\pi }{2}<s_0<+\frac{\pi }{2}\), and the following characterization is valid:

$$\begin{aligned} \Theta (s_0;X_0)=\Big \{X\in \Omega _M\Big |\, \Psi (X;X_0)\le s_0 \Big \}. \end{aligned}$$
(2.23)

3 Pseudoharmonic nonlinear combination for harmonic mappings

We refer to the covariant differentiation \(\displaystyle \,\frac{\nabla }{dt}\,\) from [13] Kap. VII, \(\S \) 5 and begin with

Definition 6

Let us consider the Riemannian metric (1.1), (1.8), (1.9) with its inverse tensor

$$\begin{aligned}&\displaystyle g^{ij}=g^{ij}(x^1,x^2)\in C^{1+\alpha }(\mathbb R^2,\mathbb R)\quad \text{ for }\quad i,j=1,2\quad \text{ satisfying }\nonumber \\&\quad \displaystyle \sum _{j=1,2}g_{ij}(x^1,x^2)\,g^{jk}(x^1,x^2)=\delta _{ik},\quad (x^1,x^2)\in \mathbb R^2\quad \text{ for }\quad i,k=1,2. \end{aligned}$$
(3.1)

Now we define the cogradient of the function \(\Psi (\,.,X_0)\) from Definition 5 as follows

$$\begin{aligned}&\displaystyle {\varvec{\nabla }}\Psi (X;X_0):=\Big (\sum _{j=1,2} g^{ij}(X)\Psi _{x^j}(X;X_0)\Big )_{i=1,2}=:\Big (f^i(X)\Big )_{i=1,2}\nonumber \\&\quad \text{ for } \text{ all }\quad X=(x^1,x^2)\in \Omega _M, \end{aligned}$$
(3.2)

where the boundary point \(\,X_0\in \partial \Omega _M\,\) is arbitrary.

Remark 6

Differentiation of the identity (3.1) yields

$$\begin{aligned}&\displaystyle -\sum _{j=1,2}\frac{\partial g_{ij}(x^1,x^2)}{\partial x^l}\,g^{jk}(x^1,x^2)= \sum _{j=1,2}g_{ij}(x^1,x^2)\,\frac{\partial g^{jk}(x^1,x^2)}{\partial x^l},\nonumber \\&\quad \text{ for } \text{ all }\quad (x^1,x^2)\in \mathbb R^2\quad \text{ and }\quad i,k,l=1,2. \end{aligned}$$
(3.3)

When an arbitrary mapping

$$\begin{aligned} X(u,v)=(x^1(u,v),x^2(u,v)):B\rightarrow \Omega _M\in C^2(B,\mathbb R^2) \end{aligned}$$
(3.4)

is given, we consider the associate auxiliary function

$$\begin{aligned} \psi (u,v):=\Psi (X(u,v);X_0)=\Psi (x^1(u,v),x^2(u,v);X_0),\quad (u,v)\in B. \end{aligned}$$
(3.5)

We immediately comprehend the following covariant chain rule

$$\begin{aligned} \displaystyle \psi _u(u,v)= & {} \Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,X_u(u,v)\Big ]_{X(u,v)},\quad (u,v)\in B,\nonumber \\ \displaystyle \psi _v(u,v)= & {} \Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,X_v(u,v)\Big ]_{X(u,v)},\quad (u,v)\in B. \end{aligned}$$
(3.6)

With the aid of the covariant product rule (see Satz 2 in [13] Kap. VII, § 5) we calculate

$$\begin{aligned} \displaystyle \psi _{uu}(u,v)= & {} \Big [\frac{\nabla \,{\varvec{\nabla }}\Psi (X(u,v);X_0)}{du},\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,\frac{\nabla \,X_u(u,v)}{du}\Big ]_{X(u,v)},\quad (u,v)\in B. \end{aligned}$$
(3.7)

Here \(\displaystyle \,\frac{\nabla }{du}\,\) denotes the covariant derivative of the respective vector field along the curve X(., v) due to Definition 1 in [13] Kap. VII, § 5. In order to evaluate the first bracket term in (3.7), we determine the partial derivative

$$\begin{aligned} \displaystyle \frac{d}{du}\Big (f^i(X(u,v))\Big )_{i=1,2}= & {} \frac{d}{du}{\varvec{\nabla }}\Psi (X(u,v);X_0)\nonumber \\= & {} \displaystyle \frac{d}{du}\Big (\sum _{j=1,2} g^{ij}(x^1(u,v),x^2(u,v))\Psi _{x^j}(x^1(u,v),x^2(u,v);X_0)\Big )_{i=1,2}\nonumber \\= & {} \displaystyle \Big (\sum _{j,k=1,2} g^{ij}(x^1(u,v),x^2(u,v))\Psi _{x^jx^k}(x^1(u,v),x^2(u,v);X_0)x^k_u\Big )_{i=1,2}\nonumber \\&\displaystyle +\Big (\sum _{j,k=1,2}\frac{\partial g^{ij}(x^1(u,v),x^2(u,v))}{\partial x^k}\Psi _{x^j}(x^1(u,v),x^2(u,v);X_0)x^k_u\Big )_{i=1,2}\nonumber \\&\text{ for } \text{ all }\quad (u,v)\in B. \end{aligned}$$
(3.8)

Now we utilize the Christoffel symbols of the first kind (see (1.6) and the formula (10) in [13] Kap. VII, § 3):

$$\begin{aligned} \displaystyle \gamma _{mjk}:= & {} \frac{1}{2} (g_{km,x^j}+g_{mj,x^k}-g_{jk,x^m})=\frac{1}{2} \sum _{i,l=1,2} g_{ml}g^{li}(g_{ki,x^j}+g_{ij,x^k}-g_{jk,x^i})\nonumber \\ \displaystyle= & {} \sum _{l=1,2}g_{ml}\Gamma _{jk}^l\quad \text{ for }\quad j,k,m=1,2. \end{aligned}$$
(3.9)

With the aid of the identities (3.8) and (3.9) and the Remark 6, we determine the first bracket term in (3.7):

$$\begin{aligned}&\displaystyle \Big [\frac{\nabla \,{\varvec{\nabla }}\Psi (X(u,v);X_0)}{du},\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\quad \displaystyle =\Bigg [\frac{\nabla \Big (f^i(X(u,v))\Big )_{i=1,2}}{du},\,\Big (x^l_u(u,v)\Big )_{l=1,2}\Bigg ]_{X(u,v)}\nonumber \\&\quad \displaystyle =\sum _{l,k=1,2} \Psi _{x^lx^k}(x^1(u,v),x^2(u,v);X_0)x^l_u x^k_u \nonumber \\&\qquad \displaystyle +\sum _{i,j,k,l=1,2} g_{li}(X(u,v))\,\frac{\partial g^{ij}(X(u,v))}{\partial x^k}\Psi _{x^j}(X(u,v);X_0)x^l_u x^k_u\nonumber \\&\qquad \displaystyle +\sum _{l,j,k=1,2} \gamma _{ljk}(X(u,v))x^l_u\,f^j(X(u,v))\,x^k_u\nonumber \\&\quad \displaystyle =\sum _{l,k=1,2} \Psi _{x^lx^k}(x^1(u,v),x^2(u,v);X_0)x^l_u x^k_u \nonumber \\&\qquad \displaystyle -\sum _{i,j,k,l=1,2}g_{li,x^k}(X(u,v))g^{ij}(X(u,v))\Psi _{x^j}(X(u,v);X_0)x^l_u x^k_u\nonumber \\&\qquad \displaystyle +\sum _{l,j,k=1,2} \gamma _{ljk}(X(u,v))x^l_u\,f^j(X(u,v))\,x^k_u\nonumber \\&\quad \displaystyle =\sum _{l,k=1,2} \Psi _{x^lx^k}(x^1(u,v),x^2(u,v);X_0)x^l_u x^k_u \nonumber \\&\qquad \displaystyle -\sum _{i,k,l=1,2}g_{li,x^k}(X(u,v))f^i(X(u,v))x^l_u x^k_u\nonumber \\&\qquad \displaystyle +\sum _{l,i,k=1,2} \gamma _{lik}(X(u,v))x^l_u\,f^i(X(u,v))\,x^k_u\nonumber \\&\quad \displaystyle =\sum _{l,k=1,2} \Psi _{x^lx^k}(x^1(u,v),x^2(u,v);X_0)x^l_u x^k_u \nonumber \\&\qquad \displaystyle +\sum _{l,i,k=1,2} \widetilde{\gamma _{lik}}(X(u,v))x^l_u\,f^i(X(u,v))\,x^k_u,\quad (u,v)\in B. \end{aligned}$$
(3.10)

Here we use the modified Christoffel symbols of the first kind

$$\begin{aligned}&\displaystyle \widetilde{\gamma _{lik}}:=\frac{1}{2} (g_{kl,x^i}-g_{li,x^k}-g_{ik,x^l})=-\frac{1}{2} (g_{li,x^k}+g_{ik,x^l}-g_{kl,x^i})=-\gamma _{ikl}\nonumber \\&\quad \text{ for }\quad i,k,l=1,2. \end{aligned}$$
(3.11)

Definition 7

We define the covariant Hessian bilinear form

$$\begin{aligned}&\displaystyle \Big [X_u(u,v),{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\quad \displaystyle :=\sum _{l,k=1,2} \Psi _{x^lx^k}(x^1(u,v),x^2(u,v);X_0)x^l_u x^k_u \nonumber \\&\qquad \displaystyle -\sum _{i,k,l=1,2} \gamma _{ikl}(X(u,v))\,f^i(X(u,v))\,x^k_u\,x^l_u,\quad (u,v)\in B. \end{aligned}$$
(3.12)

The combination of (3.7) and (3.10) – (3.12) yields the identity

$$\begin{aligned} \displaystyle \psi _{uu}(u,v)= & {} \Big [X_u(u,v),{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,\frac{\nabla \,X_u(u,v)}{du}\Big ]_{X(u,v)},\quad (u,v)\in B. \end{aligned}$$
(3.13)

Analogously, we derive the identity

$$\begin{aligned} \displaystyle \psi _{vv}(u,v)= & {} \Big [X_v(u,v),{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_v(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,\frac{\nabla \,X_v(u,v)}{dv}\Big ]_{X(u,v)},\quad (u,v)\in B. \end{aligned}$$
(3.14)

With the aid of Lemma 1 we shall see that the bilinear form in Definition 7 vanishes at each point on an appropriate one-dimensional space. More precisely, we have the

Lemma 2

(Covariant derivatives of the geodesic function)

For the geodesic function \(\Psi \) in Definition 5 the cogradient satisfies the equations

$$\begin{aligned}&\displaystyle \Big [{\varvec{\nabla }}\Psi (\mathbf Y(t,s;X_0);X_0),\mathbf Y_t(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)}=0\quad \text{ and }\nonumber \\&\quad \displaystyle \Big [{\varvec{\nabla }}\Psi (\mathbf Y(t,s;X_0);X_0),\mathbf Y_s(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)}=1\nonumber \\&\quad \displaystyle \text{ for } \text{ all }\quad 0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}. \end{aligned}$$
(3.15)

Moreover, the identity

$$\begin{aligned} {\varvec{\nabla }}\Psi (X;X_0)=\frac{\widehat{\mathbf Y_s}(X;X_0)}{{\widehat{G}}(X;X_0)},\quad X\in \Omega _M \end{aligned}$$
(3.16)

holds true. Finally, the covariant Hessian form (3.12) of the second derivatives vanishes as follows:

$$\begin{aligned}&\Big [\mathbf Y_t(t,s;X_0),{\varvec{\nabla }}^2\,\Psi (\mathbf Y(t,s;X_0);X_0),\,\mathbf Y_t(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)}=0\nonumber \\&\quad \displaystyle \text{ for } \text{ all }\quad 0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}. \end{aligned}$$
(3.17)

Here the boundary point \(\,X_0\in \partial \Omega _M\,\) is chosen arbitrarily.

Proof

  1. 1.

    We consider the auxiliary function

    $$\begin{aligned} \psi (t,s):=\Psi (\mathbf Y(t,s;X_0);X_0)=s,\quad 0< t<\tau (X_0,s),\,-\frac{\pi }{2}<s<+\frac{\pi }{2}. \end{aligned}$$
    (3.18)

    With the covariant chain rule (3.6) we determine the derivatives

    $$\begin{aligned} \displaystyle 0= & {} \psi _t(t,s)=\Big [{\varvec{\nabla }}\Psi (\mathbf Y(t,s;X_0);X_0),\,\mathbf Y_t(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)},\nonumber \\ \displaystyle 1= & {} \psi _s(t,s)=\Big [{\varvec{\nabla }}\Psi (\mathbf Y(t,s;X_0);X_0),\,\mathbf Y_s(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)},\nonumber \\&\displaystyle \text{ for } \text{ all }\quad 0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}, \end{aligned}$$
    (3.19)

    which yields the Eq. (3.15). With the aid of the Gaussian geodesic frame (2.16), we deduce the identity (3.16) from the Eq. (3.15).

  2. 2.

    Since the curve \(\mathbf Y(.,s;X_0)\) represents a geodesic, we have the identity

    $$\begin{aligned} \frac{\nabla \,\mathbf Y_t(t,s;X_0)}{dt}=0,\quad 0< t<\tau (X_0,s),\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}. \end{aligned}$$
    (3.20)

    Now the Eq. (3.13) yields

    $$\begin{aligned} \displaystyle 0= & {} \psi _{tt}(t,s)=\Big [\mathbf Y_t(t,s;X_0),{\varvec{\nabla }}^2\,\Psi (\mathbf Y(t,s;X_0);X_0),\,\mathbf Y_t(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)}\nonumber \\&\displaystyle +\Big [{\varvec{\nabla }}\Psi (\mathbf Y(t,s;X_0);X_0),\,\frac{\nabla \,\mathbf Y_t(t,s;X_0)}{dt}\Big ]_{\mathbf Y(t,s;X_0)}\nonumber \\ \displaystyle= & {} \Big [\mathbf Y_t(t,s;X_0)\,,{\varvec{\nabla }}^2\,\Psi (\mathbf Y(t,s;X_0);X_0),\,\mathbf Y_t(t,s;X_0)\Big ]_{\mathbf Y(t,s;X_0)} \nonumber \\&\displaystyle \text{ for } \text{ all }\quad 0< t<\tau (X_0,s)\,,\quad -\frac{\pi }{2}<s<+\frac{\pi }{2}\,, \end{aligned}$$
    (3.21)

    which implies the statement (3.17). \(\square \)

Now we present the principal device of our investigations within

Lemma 3

(Pseudoharmonic nonlinear combination for harmonic maps)

Let the mapping X(u, v) from (3.4) be harmonic, i.e. the Eqs. (1.5), (1.6) hold true. Then the geodesic auxiliary function \(\psi (u,v)\) in (3.5) satisfies the elliptic partial differential equation

$$\begin{aligned} \Delta \psi (u,v) + a(u,v)\psi _u(u,v)+b(u,v)\psi _v(u,v)=0\,,\quad (u,v)\in B \end{aligned}$$
(3.22)

with the continuous functions \(a=a(u,v):B\rightarrow \mathbb R\) and \(b=b(u,v):B\rightarrow \mathbb R\). The gradient \(\nabla \psi \) possesses only isolated zeroes in B and allows expansions of Hartman–Wintner-type (see Theorem 1.2 in [12] Chap. 9) there. Since this function \(\psi \) shares important properties with harmonic functions, we may address \(\psi \) as being pseudoharmonic.

Proof

  1. 1.

    The mapping X is harmonic, and we have the identity

    $$\begin{aligned} \frac{\nabla \,X_u(u,v)}{du}+ \frac{\nabla \,X_v(u,v)}{dv}=0\,,\quad (u,v)\in B. \end{aligned}$$
    (3.23)

    Now we add the Eqs. (3.13) and (3.14), and we obtain the following identity for our auxiliary function \(\psi (u,v),\,(u,v)\in B\) on account of (3.23):

    $$\begin{aligned} \displaystyle \Delta \psi (u,v)= & {} \psi _{uu}(u,v)+\psi _{vv}(u,v)\nonumber \\ \displaystyle= & {} \Big [X_u(u,v)\,,{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [X_v(u,v)\,,{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_v(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [{\varvec{\nabla }}\Psi (X(u,v);X_0),\,\frac{\nabla \,X_u(u,v)}{du}+\frac{\nabla \,X_v(u,v)}{dv} \Big ]_{X(u,v)}\nonumber \\&\displaystyle =\Big [X_u(u,v)\,,{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_u(u,v)\Big ]_{X(u,v)}\nonumber \\&\displaystyle +\Big [X_v(u,v)\,,{\varvec{\nabla }}^2\,\Psi (X(u,v);X_0),\,X_v(u,v)\Big ]_{X(u,v)}\,,\quad (u,v)\in B.\nonumber \\ \end{aligned}$$
    (3.24)
  2. 2.

    With the aid of the Gaussian geodesic frame (2.16) and the identity (3.16), we expand the vector \(X_u(u,v)\) via the covariant chain rule (3.6) as follows:

    $$\begin{aligned} \displaystyle X_u(u,v)= & {} \Big [X_u(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\&\displaystyle +\Bigg [X_u(u,v),\frac{\widehat{\mathbf Y_s}(X(u,v);X_0)}{\sqrt{{\widehat{G}}(X(u,v);X_0)}}\Bigg ]_{X(u,v)}\,\frac{\widehat{\mathbf Y_s}(X(u,v);X_0)}{\sqrt{{\widehat{G}}(X(u,v);X_0)}}\nonumber \\ \displaystyle= & {} \Big [X_u(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\&\displaystyle +\Bigg [X_u(u,v),\frac{\widehat{\mathbf Y_s}(X(u,v);X_0)}{{\widehat{G}}(X(u,v);X_0)}\Bigg ]_{X(u,v)}\,\widehat{\mathbf Y_s}(X(u,v);X_0)\nonumber \\ \displaystyle= & {} \Big [X_u(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\ \displaystyle&+\Big [X_u(u,v),{\varvec{\nabla }}\Psi (X(u,v);X_0)\Big ]_{X(u,v)}\,\widehat{\mathbf Y_s}(X(u,v);X_0)\nonumber \\ \displaystyle= & {} \Big [X_u(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\&\displaystyle +\psi _u(u,v)\,\widehat{\mathbf Y_s}(X(u,v);X_0)\,,\quad (u,v)\in B. \end{aligned}$$
    (3.25)

    Proceeding in the same way for the derivative with respect to v, we arrive at the following equations:

    $$\begin{aligned} \displaystyle X_u(u,v)= & {} \Big [X_u(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\&\displaystyle +\psi _u(u,v)\,\widehat{\mathbf Y_s}(X(u,v);X_0)\,,\quad (u,v)\in B\,;\nonumber \\ \displaystyle X_v(u,v)= & {} \Big [X_v(u,v),\widehat{\mathbf Y_t}(X(u,v);X_0)\Big ]_{X(u,v)}\widehat{\mathbf Y_t}(X(u,v);X_0)\nonumber \\&\displaystyle +\psi _v(u,v)\,\widehat{\mathbf Y_s}(X(u,v);X_0)\,,\quad (u,v)\in B. \end{aligned}$$
    (3.26)
  3. 3.

    When we insert the vectors \(X_u(u,v)\) and \(X_v(u,v)\) from (3.26) into the covariant Hessian forms within (3.24) and observe the property (3.17), we receive the representation (3.22) with continuous coefficient functions. \(\square \)

Remark 7

Similar arguments for the Euclidean situation under semi-free boundary conditions have been established in [5] Proposition 3 within my joint investigation with Hildebrandt.

4 Convex hull property, univalency and transversality for harmonic mappings

We start with the central definition and assume that setting throughout this section.

Definition 8

Let \(ds^2\) denote a stable Riemannian metric (1.1), (1.8), (1.9) on the disc \(\Omega _M\) of radius \(0<M<+\infty \) with a moderate deviation (1.10) by the constant \(\displaystyle a\in (0,\frac{1}{2M})\) from the Euclidean metric. For each continuous boundary function

$$\begin{aligned} \Phi \in C^{0}(\partial B,\mathbb R^2)\quad \text{ with }\quad |\Phi (u,v)|\le M,\,\forall \,(u,v)\in \partial B \end{aligned}$$

we call the function

$$\begin{aligned} \displaystyle X= & {} X(u,v)=(x^1(u,v),x^2(u,v)):{\overline{B}}\rightarrow \mathbb R^2\in C^2(B,\mathbb R^2)\cap C^0({\overline{B}},\mathbb R^2)\nonumber \\&\displaystyle \text{ with }\quad |X(u,v)|\le M\quad \text{ for } \text{ all }\quad (u,v)\in {\overline{B}} \end{aligned}$$
(4.1)

a solution of the Dirichlet problem \(\,{\mathcal {P}} (\Omega _M,ds^2;\Phi )\,\), when the function X satisfies the system (1.5), (1.6) of harmonic mappings and fulfills the boundary condition

$$\begin{aligned} X(u,v)=\Phi (u,v)\quad \text{ for } \text{ all }\quad (u,v)\in \partial B. \end{aligned}$$
(4.2)

Theorem 3

(Convex hull property for harmonic mappings)

Let the continuous function \(\Phi :\partial B \rightarrow \Omega _M\in C^0(\partial B)\) with the boundary point set \(\displaystyle F:=\Phi (\partial B)\subset \Omega _M\) and its convex hull \(\displaystyle {{\mathcal {H}}}(F)\subset \Omega _M\) due to Definition 4 be given. For each solution

$$\begin{aligned} X=X(u,v)=(x^1(u,v),x^2(u,v))\in {\mathcal {P}} (\Omega _M,ds^2;\Phi ) \end{aligned}$$

of the Dirichlet problem we have the following inclusion:

$$\begin{aligned} X(u,v)\in {{\mathcal {H}}}(F) \quad \text{ for } \text{ all }\quad (u,v)\in {\overline{B}}. \end{aligned}$$
(4.3)

Proof

  1. 1.

    The boundary point set \(F:=\Phi (\partial B)\subset \Omega _M\) is compact in \(\Omega _M\), and the convex hull of the boundary values \({{\mathcal {H}}}(F)\subset \Omega _M\) as well. Therefore, we can find a unique number \(\displaystyle \,\sigma (X_0,F)\in (-\frac{\pi }{2},+\frac{\pi }{2})\,\), such that

    $$\begin{aligned}&\displaystyle \bigcap \Big \{\Theta (s_0;X_0)\Big |\,s_0\in (-\frac{\pi }{2},+\frac{\pi }{2}):\,F\subset \Theta (s_0;X_0)\Big \} =\Theta (\sigma (X_0,F);X_0)\nonumber \\&\quad =\Big \{X\in \Omega _M\Big |\,\Psi (X;X_0)\le \sigma (X_0,F)\Big \}\quad \text{ for } \text{ each } \text{ point }\quad X_0\in \partial \Omega _M. \end{aligned}$$
    (4.4)

    Here we have utilized the characterization (2.23) for the last identity. Now we determine the convex hull of the boundary point set as follows:

    $$\begin{aligned}&\displaystyle {\mathcal {H}}(F)=\bigcap \Big \{\Theta (s_0;X_0)\Big |\,X_0\in \partial \Omega _M,\,s_0\in (-\frac{\pi }{2},+\frac{\pi }{2}):\,F\subset \Theta (s_0;X_0)\Big \}\nonumber \\&\quad \displaystyle =\bigcap _{X_0\in \partial \Omega _M} \Theta (\sigma (X_0,F);X_0)= \bigcap _{X_0\in \partial \Omega _M}\Big \{X\in \Omega _M\Big |\,\Psi (X;X_0)\le \sigma (X_0,F)\Big \}.\nonumber \\ \end{aligned}$$
    (4.5)
  2. 2.

    With the aid of the geometric maximum principle by E. Heinz, we can see as in Remark 2 that the inclusion \(X(\partial B)\subset \Omega _M\) implies the property \(X(B)\subset \Omega _M\). For arbitrary points \(X_0\in \partial \Omega _M\) we consider the geodesic auxiliary function

    $$\begin{aligned} \psi (u,v):=\Psi (X(u,v);X_0)\,,\quad (u,v)\in {\overline{B}}. \end{aligned}$$
    (4.6)

    Since the inclusion \(F\subset \Theta (\sigma (X_0,F);X_0)\) for all \(X_0\in \partial \Omega _M\) holds true, we receive

    $$\begin{aligned} \psi (u,v)\le \sigma (X_0,F)\quad \text{ for } \text{ all }\quad (u,v)\in \partial B. \end{aligned}$$
    (4.7)

    Now Lemma 3 implies that the function \(\psi \) is subject to the maximum principle, which gives us the following statement:

    $$\begin{aligned} \Psi (X(u,v);X_0)\le \sigma (X_0,F)\,,\quad (u,v)\in {\overline{B}}\quad \text{ for } \text{ all } \text{ points }\quad X_0\in \partial \Omega _M. \end{aligned}$$
    (4.8)

    On account of (4.5), we obtain that \(\displaystyle \,X({\overline{B}})\subset {\mathcal {H}}(F)\,\) holds true. \(\square \)

Definition 9

A Jordan contour \(\Gamma \subset \Omega _M\) is called convex in \(\Omega _M\), when the following properties are fulfilled:

  1. (i)

    The Jordan contour \(\Gamma \) coincides with the boundary \(\partial {\mathcal {H}}(\Gamma )\) of its convex hull, and the interior \(I(\Gamma )\) of the contour \(\Gamma \) corresponds to the open kernel of the convex hull \({\mathcal {H}}(\Gamma )\).

  2. (ii)

    A geodesic \(\mathbf Y(t,s_0;X_0),\,0\le t\le \tau (X_0,s_0)\) for the parameter \(\displaystyle s_0\in (-\frac{\pi }{2},+\frac{\pi }{2})\), such that \(\mathbf Y(.,s_0;X_0)\) meets the interior \(I(\Gamma )\) at an inner point \(Y_0\in I(\Gamma )\), shall decompose the Jordan curve into the closed Jordan arcs

    $$\begin{aligned} \Gamma ^-(X_0,s_0):=\Gamma \cap \Theta (s_0,X_0)\quad \text{ and }\quad \Gamma ^+(X_0,s_0):=\overline{\Gamma {\setminus }\Theta (s_0,X_0)}. \end{aligned}$$

    These arcs meet at their end points on the geodesic \(\mathbf Y(.,s_0;X_0)\) above.

With the original method by Kneser [9] for the Euclidean plane, which we adapt to the Riemannian situation here, we shall establish the subsequent

Theorem 4

(Univalency for harmonic mappings)

Let the convex Jordan contour \(\Gamma \subset \Omega _M\) and the topological boundary function \(\Phi :\partial B\rightarrow \Gamma \in C^0(\partial B,\mathbb R^2)\) be given. Then each solution

$$\begin{aligned} X=X(u,v)=(x^1(u,v),x^2(u,v))\in {\mathcal {P}} (\Omega _M,ds^2;\Phi ) \end{aligned}$$

of the Dirichlet problem furnishes a topological mapping of \({\overline{B}}\) onto \(\overline{I(\Gamma )}\) and a \(C^2\)-diffeomorphism of B onto \(I(\Gamma )\).

Proof

  1. 1.

    From Theorem 3 and Definition 9, (i) we infer the inclusion

    $$\begin{aligned} X({\overline{B}})\subset {\mathcal {H}}(\Gamma )=I(\Gamma )\cup \Gamma . \end{aligned}$$
    (4.9)

    Moreover, the strict inclusion

    $$\begin{aligned} X(B)\subset I(\Gamma ) \end{aligned}$$
    (4.10)

    is valid, which we deduce as follows: If the statement (4.10) were violated, there exists a point \((u_0,v_0)\in B\) with \(Y_0=X(u_0,v_0)\in \Gamma \). On account of Definition 9 we can find a point \(X_0\in \partial \Omega _M\) and a value \(\displaystyle \,-\frac{\pi }{2}<s_0<+\frac{\pi }{2}\,\), such that

    $$\begin{aligned} \Gamma \subset \Theta (s_0;X_0)\quad \text{ and }\quad Y_0\in \Gamma \cap \partial \Theta (s_0;X_0) \end{aligned}$$
    (4.11)

    holds true. Now we consider the auxiliary function

    $$\begin{aligned} \psi (u,v):=\Psi (X(u,v);X_0)\le s_0,\,(u,v)\in {\overline{B}}\,\text{ with }\,\psi (u_0,v_0)=\Psi (Y_0;X_0)=s_0.\nonumber \\ \end{aligned}$$
    (4.12)

    From Lemma 3 we see that \(\psi \) is a pseudoharmonic function and cannot attain a strict maximum within B. Consequently, the equality

    $$\begin{aligned} \psi (u,v)=s_0\quad \text{ for } \text{ all }\quad (u,v)\in {\overline{B}} \end{aligned}$$
    (4.13)

    holds true, which yields an evident contradiction. Therefore, the strict inclusion (4.10) is valid.

  2. 2.

    Now we show indirectly that the Jacobian of the mapping X does not vanish:

    $$\begin{aligned} J_X(u,v):=\frac{\partial (x^1(u,v),x^2(u,v))}{\partial (\,u\,,\,v\,)} =\left| \begin{array}{c} x^1_u(u,v),x^1_v(u,v)\\ x^2_u(u,v),x^2_v(u,v)\end{array}\right| \ne 0\,,\forall \,(u,v)\in B.\nonumber \\ \end{aligned}$$
    (4.14)

    If the statement (4.14) were violated, there exists a point

    $$\begin{aligned} (u_0,v_0)\in B\quad \text{ with }\quad Y_0:=X(u_0,v_0)\in I(\Gamma )\,, \end{aligned}$$

    such that the vectors \(\{X_u(u_0,v_0),\,X_v(u_0,v_0)\}\) are linearly dependent. Consequently, we find a unit vector \(Z_0\) orthogonal to these vectors as follows:

    $$\begin{aligned}&\displaystyle Z_0\in \mathbb R^2{\setminus }\{(0,0)\}\quad \text{ with }\quad \Big [Z_0\,,Z_0\Big ]_{X(u_0,v_0)}=1\,,\nonumber \\&\quad \displaystyle \Big [Z_0\,,X_u(u_0,v_0)\Big ]_{X(u_0,v_0)}=0=\Big [Z_0\,,X_v(u_0,v_0)\Big ]_{X(u_0,v_0)}. \end{aligned}$$
    (4.15)
  3. 3.

    Now the Gaussian geodesic frame at the fixed point \(Y_0\in \Omega _M\)

    $$\begin{aligned} \Bigg \{ \widehat{\mathbf Y_t}(Y_0;X_0)\,,\,\frac{\widehat{\mathbf Y_s}(Y_0;X_0)}{\sqrt{{\widehat{G}}(Y_0;X_0)}}\Bigg \}\,,\quad X_0\in \partial \Omega _M \end{aligned}$$
    (4.16)

    performs one positive-oriented and continuous rotation, when \(X_0\) traverses the circumference \(\partial \Omega _M\) once in positive orientation. This results from the construction of the geodesic vector fields, which depend continuously on the point \(X_0\in \partial \Omega _M\) together with their nonvanishing derivatives. Therefore, we can choose a point \(X_0\in \partial \Omega _M\) such that

    $$\begin{aligned} \frac{\widehat{\mathbf Y_s}(Y_0;X_0)}{\sqrt{{\widehat{G}}(Y_0;X_0)}}=Z_0 \end{aligned}$$
    (4.17)

    holds true. With the aid of (3.16) and (4.17), we obtain the following representation for the cogradient of the geodesic function \(\Psi \)

    $$\begin{aligned} {\varvec{\nabla }}\Psi (Y_0;X_0)=\frac{\widehat{\mathbf Y_s}(Y_0;X_0)}{{\widehat{G}}(Y_0;X_0)}=\lambda \,Z_0\quad \text{ with }\quad \lambda :=\frac{1}{\sqrt{{\widehat{G}}(Y_0;X_0)}}. \end{aligned}$$
    (4.18)
  4. 4.

    Let us now consider the geodesic auxiliary function

    $$\begin{aligned} \psi (u,v):=\Psi (X(u,v);X_0)\,,\quad (u,v)\in {\overline{B}}. \end{aligned}$$
    (4.19)

    With the aid of (4.15) and (4.18) we derive

    $$\begin{aligned} \displaystyle \psi _u(u_0,v_0)= & {} \Big [{\varvec{\nabla }}\Psi (Y_0;X_0),\,X_u(u_0,v_0)\Big ]_{{X(u_0,v_0)}}\nonumber \\ \displaystyle= & {} \lambda \,\Big [Z_0,X_u(u_0,v_0)\Big ]_{{X(u_0,v_0)}}=0\,;\nonumber \\ \displaystyle \psi _v(u_0,v_0)= & {} \Big [{\varvec{\nabla }}\Psi (Y_0;X_0),\,X_v(u_0,v_0)\Big ]_{{X(u_0,v_0)}}\nonumber \\ \displaystyle= & {} \lambda \,\Big [Z_0,X_v(u_0,v_0)\Big ]_{{X(u_0,v_0)}}=0. \end{aligned}$$
    (4.20)

    Since the function \(\psi \) is pseudoharmonic due to Lemma 3 and \(\,\nabla \psi (u_0,v_0)=(0,0)\,\) holds true, now \(\psi \) represents a saddle point near \((u_0,v_0)\). This behavior propagates to the boundary \(\partial B\) on account of the maximum/minimum principle. This yields a contradiction to the behavior of the function \(\psi :\partial B\rightarrow \mathbb R\) on the boundary, which only possesses two points for the level \(s_0\) due to Definition 9, (ii) Consequently, the Jacobian \(J_X\) is not allowed to vanish within B, and the statement (4.14) holds true. For an exact proof, we can follow the arguments for harmonic functions in Lemma 2 and Lemma 3 of our book on Minimal Surfaces [2] within Section 4.9. These arguments remain valid for the pseudoharmonic function \(\psi \), due to the asymptotic expansions of P. Hartman and A. Wintner (see Theorem 1.2 in [12] Chap. 9.) at their critical points.

  5. 5.

    With the monodromy principle (see Lemma 1 in [2], Sect. 4.9) we can infer the topological character of the mapping

    $$\begin{aligned} X:{\overline{B}}\rightarrow \overline{I(\Gamma )}\subset \Omega _M \end{aligned}$$

    from (4.14) and the property that the boundary representation \(X:\partial B\rightarrow \Gamma \) is topological. Alternatively, we can use an index-argument from [11] Hilfssatz 7 in order to show that the mapping \(\,X:{\overline{B}}\rightarrow \overline{I(\Gamma )}\,\) is one-to-one. \(\square \)

Remark 8

In the Euclidean situation, we find this result by T. Radó and H. Kneser in § 398 of J. C. C. Nitsche’s monograph [10] Vorlesungen über Minimalflächen.

Furthermore, we refer to Proposition 4.2 in my joint treatise [4] with S. Hildebrandt.

The following statement contains the transversality of harmonic mappings to the boundary. More precisely, we shall establish

Theorem 5

(Existence of \(C^{2+\alpha }({\overline{B}},\overline{\Omega _M})\)-diffeomorphisms for \({{\mathcal {P}}}(\Omega _M,ds^2;\Phi )\)) Let the \(C^{2+\alpha }(\partial B,\partial \Omega _M)\)-diffeomorphic boundary function \(\Phi :\partial B \rightarrow \partial \Omega _M\) be given. Then there exists a \(C^{2+\alpha }({\overline{B}},\overline{\Omega _M})\)-diffeomorphism

$$\begin{aligned} X=X(u,v)=(x^1(u,v),x^2(u,v)):{\overline{B}}\rightarrow \overline{\Omega }_M\,, \end{aligned}$$

which furnishes a solution of the Dirichlet Problem \({{\mathcal {P}}}(\Omega _M,ds^2;\Phi )\).

Proof

  1. 1.

    We build upon our exsistence result in Theorem 2, and we receive a solution \(X=X(u,v)\in C^{2+\alpha }({\overline{B}},\overline{\Omega _M})\) for the Dirichlet problem \({{\mathcal {P}}}(\Omega _M,ds^2;\Phi )\). By the geometric maximum principle of E. Heinz the function

    $$\begin{aligned} \chi (u,v):=|X(u,v)|^2\,,\quad (u,v)\in {\overline{B}}\quad \text{ satisfies }\quad \Delta \chi (u,v)\ge 0\,,\quad (u,v)\in B. \end{aligned}$$

    The boundary point lemma of E. Hopf implies the following inequality for the derivative w. r. t. the exterior normal \(\nu \) to B:

    $$\begin{aligned} 0<\frac{d}{d\nu }\chi (u_1,v_1)=2\,X(u_1,v_1) \cdot \frac{d}{d\nu } X(u_1,v_1)\quad \text{ for } \text{ all } \text{ points }\quad (u_1,v_1)\in \partial B.\nonumber \\ \end{aligned}$$
    (4.21)

    This property (4.21) together with the arguments in [11] Satz 2 yield that our mapping X is transversal in the following sense:

    $$\begin{aligned} J_X(u,v)\ne 0\quad \text{ for } \text{ all }\quad (u,v)\in \partial B. \end{aligned}$$
    (4.22)
  2. 2.

    Now we follow the parts (2)–(4) in the proof of Theorem 4, in order to exclude zeroes of the Jacobian \(J_X\) within B. When the geodesic field \(\mathbf Y(t,s;X_0)\) has the center \(X_0\in \partial \Omega _M\), we exempt from \(\Omega _M\) a disc about this singularity for a sufficiently small number \(\epsilon >0\). With the domain

    $$\begin{aligned} \Omega _M^{\,\epsilon }(X_0):=\Big \{X\in \Omega _M\Big |\,|X-X_0|>\epsilon \Big \} \end{aligned}$$

    we modify the arguments in part (4) within the proof of Theorem 4, and we consider alternatively the auxiliary function

    $$\begin{aligned} \psi (u,v):=\Psi (X(u,v);X_0),(u,v)\in {\overline{B}}_{\epsilon }:=\Big \{(u,v)\in {\overline{B}}\Big |X(u,v)\in \overline{\Omega _M^{\,\epsilon }(X_0)}\Big \}.\nonumber \\ \end{aligned}$$
    (4.23)

    Thus we can exclude each zero of the Jacobian in the interior of the disc B. With the part (5) in the proof of Theorem 4, we complete the derivation of our result above. \(\square \)

Remark 9

In order to show that a conformally parametrized H-surface represents a graph, one has to prove that the associate plane mapping is one-to-one. Here the investigation [11] contains as the decisive step that transversal mappings yield necessarily a diffeomorphism. There we need a stability condition in the sense that the second variation of the associate parametric integral is nonnegative.

5 Harmonic embeddings within the hemisphere

For all radii \(0<M<+\infty \) with their associate discs

$$\begin{aligned} \Omega _M:=\Big \{X=(x^1,x^2)\in \mathbb R^2:\,|X|<M\Big \} \end{aligned}$$

we consider the upper hemisphere \(S^+_M\) in the following representation

$$\begin{aligned} Z(x^1,x^2):=\Big (x^1,x^2,\sqrt{M^2-|X|^2}\Big )\,,\quad X=(x^1,x^2)\in \Omega _M. \end{aligned}$$
(5.1)

Then we derive

$$\begin{aligned} Z_{x^i}(x^1,x^2)=\Big (\delta _{1i},\delta _{2i},\frac{-\,x^i}{\sqrt{M^2-|X|^2}}\Big )\,,\quad X=(x^1,x^2)\in \Omega _M,\, i=1,2 \end{aligned}$$
(5.2)

and determine their first fundamental form (1.1) as follows

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle g_{ij}:=Z_{x^i}\cdot Z_{x^j}(x^1,x^2)=\delta _{1i}\delta _{1j}+\delta _{2i}\delta _{2j}+\frac{x^i x^j}{M^2-|X|^2}=\delta _{ij}+\frac{x^i x^j}{M^2-|X|^2}\\ &{}\quad \displaystyle \text{ for } \text{ all }\quad X=(x^1,x^2)\in \Omega _M\quad \text{ and }\quad i,j=1,2. \end{array} \end{aligned}$$
(5.3)

We denote the hemispherical metric (1.1), (5.3) by \(\,ds^2(M)\,\). This metric becomes singular near the boundary \(\partial \Omega _M\), and our Theorem 2 is not applicable globally. S. Hildebrandt, H. Kaul and K. Widman have constructed harmonic mappings into complete Riemannian manifolds with positive sectional curvature by direct variational methods (see [3]). Since this result is especially valid for hemispheres, we receive the following

Theorem 6

(Dirichlet problem for hemispherical harmonic mappings) Let a radius \(0<M<+\infty \) be chosen arbitrarily. For each boundary function

$$\begin{aligned} \Phi \in C^0(\partial B,\Omega _M)\quad \text{ possessing } \text{ a }\quad W^{1,2}(B,\mathbb R^2)-\text{ extension } \end{aligned}$$
(5.4)

there exists a solution

$$\begin{aligned} X=X(u,v)=(x^1(u,v),x^2(u,v)):{\overline{B}}\rightarrow \Omega _M\in C^{2+\alpha }(B,\mathbb R^2)\cap C^0({\overline{B}},\mathbb R^2) \end{aligned}$$
(5.5)

for the Dirichlet problem \({\mathcal {P}}(\Omega _M,ds^2(M);\Phi )\) of the harmonic mapping associated with the hemispherical metric \(\,ds^2(M)\,\).

Proof

See the Theorems 1–4 in [3]. \(\square \)

We construct a field of geodesics, which emanates from an arbitrary equatorial point

$$\begin{aligned} Z_{\vartheta }=\Big (M\cos \vartheta ,M\sin \vartheta ,\,0\Big )\in \partial S^+_M\,,\quad 0\le \vartheta \le 2\pi \end{aligned}$$
(5.6)

and simply covers the hemisphere. We begin with the great circle on \(S^+_M\)

$$\begin{aligned} \Big (M\cos \Big (\frac{t}{M}\Big ),\,0\,,M\sin \Big (\frac{t}{M}\Big )\Big )^*\,,\quad 0<t<M\pi . \end{aligned}$$
(5.7)

This circle represents a geodesic without interior conjugate points; it starts at the point \(Z_0=(M,0,0)\) and ends at the antipodal point \(Z_{\pi }=(-M,0,0)\), which is conjugate to \(Z_{0}\). We use the rotation by the angle s about the \(x^1\)-axis

$$\begin{aligned} D^1_s:=\left( \begin{array}{lll} 1 &{}\quad 0&{} \quad 0\\ 0&{}\quad \cos s&{} \quad -\sin s\\ 0&{}\quad \sin s&{}\quad \cos s \end{array}\right) ,\quad -\frac{\pi }{2}<s<+\frac{\pi }{2} \end{aligned}$$
(5.8)

and the rotation by the angle \(\vartheta \) about the \(x^3\)-axis

$$\begin{aligned} D^3_{\vartheta }:=\left( \begin{array}{rrr} \cos \vartheta &{} \quad -\sin \vartheta &{} \quad 0\\ \sin \vartheta &{} \quad \cos \vartheta &{} \quad 0\\ 0&{} \quad 0&{} \quad 1\end{array}\right) ,\quad 0\le \vartheta \le 2 \pi . \end{aligned}$$
(5.9)

We obtain the field of geodesics

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle \mathbf Z(t,s;\vartheta ):=D^3_{\vartheta }\circ D^1_s\circ \Big (M\cos \Big (\frac{t}{M}\Big ),\,0\,,M\sin \Big (\frac{t}{M}\Big )\Big )^*\,,\\ &{}\quad \displaystyle 0<t<M\pi ,\,-\frac{\pi }{2}<s<+\frac{\pi }{2}\quad \text{ for } \text{ all } \text{ angles }\quad 0\le \vartheta \le 2 \pi . \end{array} \end{aligned}$$
(5.10)

Via the projection from the Euclidean space onto the plane

$$\begin{aligned} \Pi ^3(Z)=\Pi ^3(x^1,x^2,x^3):=(x^1,x^2)=X\in \mathbb R^2\,,\quad Z=(x^1,x^2,x^3)\in \mathbb R^3 \end{aligned}$$
(5.11)

we see from the construction above that the family of functions

$$\begin{aligned} \begin{array}{c} \displaystyle \mathbf Y(t,s;\vartheta ):=\Pi ^3\circ \mathbf Z(t,s;\vartheta )\,,\quad 0<t<M\pi ,\,-\frac{\pi }{2}<s<+\frac{\pi }{2} \end{array} \end{aligned}$$
(5.12)

constitutes a central field of geodesics for the hemispherical metric (1.1), (5.3). This central field \(\mathbf Y(.,.;\vartheta )\) simply covers the disc \(\Omega _M\) and emanates from the singular point \(\,X_0=\Pi ^3(X_{\vartheta })\in \partial \Omega _M\,\), where \(\,0\le \vartheta \le 2 \pi \,\) denotes an arbitrary angle.

With the aid of this central field of geodesics, we introduce geodesic regions and convex hulls for compact sets within \(\Omega _M\) as in Definition 4. Furthermore, we can define the geodesic function parallel to Definition 5 and receive the fundamental Lemma 3 for the hemispherical metric \(\,ds^2(M)\,\). Finally, we characterize convex Jordan contours \(\Gamma \subset \Omega _M\) as in Definition 9 with respect of the hemispherical metric. By the arguments in the proofs for Theorem 3 and Theorem 4, we can establish

Theorem 7

(Harmonic embeddings within the hemisphere) Let the convex Jordan contour \(\Gamma \subset \Omega _M\) and the topological boundary function \(\Phi :\partial B\rightarrow \Gamma \) as in (5.4) be given. Then each solution \(\,X(u,v)=(x^1(u,v),x^2(u,v))\,\) of the regularity (5.5) for the Dirichlet problem \({\mathcal {P}}(\Omega _M,ds^2(M);\Phi )\) of the hemispherical metric \(ds^2(M)\) furnishes a topological mapping of \({\overline{B}}\) onto \(\overline{I(\Gamma )}\) and a \(C^{2+\alpha }\)-diffeomorphism of B onto \(I(\Gamma )\).

Remark 10

Let \({\mathcal {M}}\) denote a 2-dimensional, geodesically complete, connected and oriented Riemannian manifold of the class \(C^3\) without boundary, whose Gaussian curvature is bounded from above by the constant \(\kappa \in [0,+\infty )\) as follows:

$$\begin{aligned} K(X)\le \kappa \quad \text{ for } \text{ all }\quad X\in {\mathcal {M}}. \end{aligned}$$
(5.13)

On this manifold we choose an arbitrary point \(P\in {\mathcal {M}}\) and a radius \(\displaystyle \,0<M<\frac{\pi }{2\sqrt{\kappa +}}\,\) such that the geodesic disc

$$\begin{aligned} {\mathcal {B}}_M(P):=\Big \{Q\in {\mathcal {M}}\Big |\,\text{ dist }(Q,P)\le M\Big \} \end{aligned}$$
(5.14)

satisfies a cut-locus-condition (see the treatise [3] by S. Hildebrandt, H. Kaul and K. Widman). Then we can solve the Dirichlet problem for harmonic mappings in the interior of \({\mathcal {B}}_M(P)\) by these investigations using direct variational methods. Here we can apply our methods from above in order to obtain harmonic diffeomorphisms.

Example 1

Harmonic diffeomorphisms in the Poincaré half-plane.

We consider the Poincaré half-plane (see [8] 5.1.3 in the Lectures by W. Klingenberg) with the following coefficients for their first fundamental form \(ds^2\) from (1.1):

$$\begin{aligned} \begin{array}{c} \displaystyle g_{ij}(X):=\frac{1}{x_2^{\,2}}\,\delta _{ij},\,X\in \mathbb R^2_+:=\Big \{X=(x^1,x^2)\in \mathbb R^2\Big |\,x_2>0\Big \}\quad \text{ for }\,i,j=1,2. \end{array} \end{aligned}$$
(5.15)

Due to [8] Satz 5.1.7, the geodesics in the Poincaré half-plane with their Gaussian curvature \(\,K \equiv -1\,\) consist of all circular arcs within \(\mathbb R^2_+\) meeting the \(x_1\)-axis perpendicularly and the rays emanating orthogonally from the \(x_1\)-axis, which we address as orthocircles. From [1] §§ 81–84 we see that the geodesic discs \({\mathcal {B}}_M(P)\subset \mathbb R^2_+\) with their center P on the positive \(x_2\)-axis possess a convex circumference

$$\begin{aligned} \partial {\mathcal {B}}_M(P):=\Big \{Q\in {\mathcal {M}}\Big |\,\text{ dist }(Q,P)= M\Big \} \end{aligned}$$
(5.16)

with the constant geodesic curvature

$$\begin{aligned} \kappa _g(X)>0\quad \text{ for } \text{ all }\quad X\in \partial {\mathcal {B}}_M(P). \end{aligned}$$
(5.17)

Due to Figure 14 in [1] § 84 of the Grundlehren by W. Blaschke and K. Leichtweiß the circumferences for these geodesic discs constitute the orthogonal trajectories of the orthocircles. The geodesic discs \({\mathcal {B}}_M(P_M)\) exhaust the Poincaré half-plane for \(M\rightarrow +\infty \). Here we also refer to Abb. 5.1 in [8] 5.1.

Each boundary point \(X_0\in \partial {\mathcal {B}}_M(P_M)\) possesses a central field of geodesics, which emanates from \(X_0\) and foliates \({\mathcal {B}}_M(P_M)\). Consequently, the variational solution \(X(u,v),\,(u,v)\in {\overline{B}}\) of the Dirichlet problem for harmonic mappings by Hildebrandt, Kaul and Widman [3] exists within the geodesic discs of all radii \(M>0\). Then we can apply the methods from Sects. 2 to 4 above, and we see that this solution X shares the convex-hull property. Furthermore, this variational solution X yields a diffeomorphism in B and a topological mapping on \({\overline{B}}\) for topological boundary representations onto convex Jordan contours \(\Gamma \), which are contained in the interior of the disc \({\mathcal {B}}_M(P_M)\). Thus we receive an analogue of Theorem 7 within the Poincaré half-plane.