Let $S$ be a minimal surface in $R^3$ space and $\Gamma$ a family of curves on $S$. Further let $L$ be a plane in $R^3$ and
$\Gamma'$ the family which is the projection of $\Gamma$ onto $L$.
Are there some results related $Mod(\Gamma)$ and $Mod(\Gamma')$?
Vladimir Miklyukov's book "Conformal Maps of Nonsmooth Surfaces and Their Applications'" has some related estimates. They have been used, for example, by Allen Weitzman for studying harmonic mappings and minimal surfaces. This is certainly a very interesting topic. Unfortunately, Miklyukov passed away already, so we can't discuss with him on this.
Dear Antti, thanks.
I try to use realation between $Mod(\Gamma)$ and $Mod(\Gamma')$ in connection
with the following question which is appeared in communication with Franc Forstneric.
By $\mathbb{D}$ and $B_3$ we denote the unit disk and the unit ball in $R^3$ respectively.
For $p\in \mathbb{B_3} $, let $CH(p)$ denote the family of all harmonics maps $f$ from $\mathbb{D}$ into the unit ball $B_3$ with $f(0)=p$.
Set $L(p)= \sup \{ |f'(0)|: f \in CH(p) \}$ and $K(p)=\frac{L(p)}{1-|p|^2}$, where $|f'(0)|$ denotes the norm of the liner operator $f'(0)$.
Whether $K(p)=1$?
Do you mean that the mapping is into the unit ball of R3 (i.e. some surface that is contained in the unit ball), or into the surface of the unit ball (the unit sphere)?
The mapping is into the unit ball of R3 (i.e. some surface that is contained in the unit ball. The question is how to obtain the optimal Schwarz lemma for conformal
minimal disks in domains in $\mathbb{R}^n$.
Can we use the Gauss map definition: A surface $M\subset \mathbb{R}^3$ is minimal if and only if its stereographically projected Gauss map $g: M \rightarrow \mathbb{C} \cup \{\infty\}$ is meromorphic with respect to the underlying Riemann surface structure, and $M$ is not a piece of a sphere.
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