Suppose you have a function $f$ on hyperbolic $3$ dimensional space $H^3$ which is harmonic with respect to the hyperbolic Laplacian, where $H^3$ is thought of as the open $3$-ball $B^3$ in $\mathbb{R}^3$, such that $f$ extends continuously to the closure $\operatorname{cl}(B^3)$, in such a way that the restriction of (the extension of) $f$ to the sphere at infinity $S^2$ is holomorphic, when $S^2$ is thought of as the Riemann sphere via stereographic projection.
Is there some kind of maximal/minimal principle for $f$ in terms of its restriction to the sphere at infinity?
I guess one may ask a more general question for $f$ defined on $H^{n+1}$ which is (hyperbolic) harmonic, extends continuously to the closure of $B^{n+1}$, in such a way that $f$ is conformal when restricted to the $n$-sphere at infinity. Is there some kind of maximal/minimual principle for $f$ in such a context?
Edit 1: to be more precise, since holomorphic functions on $S^2$ are constant, I actually require the function $f$ to be a section of a complex line bundle $L$ defined on $\operatorname{cl}(B)$, which is a holomorphic bundle when restricted to the $2$-sphere at infinity (in my less general setting). $L$ may have more structure perhaps, such as a connection, satisfying some conditions. Anyway, I wonder if this setup rings a bell, i.e. has been studied before.
Edit 2: upon thinking about this post a bit more, I think that it is not a good idea to require $f$ to be harmonic on all of harmonic space. Let me modify my question a little bit. If $G(x,y)$ is a Green's function of a second order elliptic linear differential operator on a manifold $M$ with boundary $\partial M$, and if $G(x,y)$ is required to vanish when either $x$ or $y$ is on $\partial M$, then is it true that $G(x,y)$ is positive in the "bulk", i.e. in the interior of $M$? I think this makes sense from a physical point of view. It would be strange if $G(x,y)$ became negative for some $x$, $y$ in the bulk, $x \neq y$, since it is the value of the potential at $x$ created by a point source at $y$. Perhaps one may apply a strong version of the maximum principle to prove that?