Geometric/Intuitive Interpretation of Schwarz Lemma I refer to Schwarz Lemma (https://en.wikipedia.org/wiki/Schwarz_lemma).
Is there a geometrical or intuitive meaning of Schwarz Lemma that helps to understand or remember this result better?
Thanks for any help.
 A: The geometric interpretation of the Schwarz lemma is that a holomorphic map of the unit disk into the unit disk is distance nonincreasing with respect to the hyperbolic metric on the unit disk. This interpretation is due to Ahlfors. See the survey of R. Osserman for references. S. T. Yau later generalized this to the context of biholomorphic maps from a complete Kähler manifold with Ricci curvature bounded from below to a Hermitian manifold with biholomorphic sectional curvature bounded from above by a negative constant. A proof of a more general result can be found in this article of V. Tosatti.
A: If $f:D→D$ be analytic and $f(0)=0$ then (a)$|f(z)|$$\le |z|$ and (b)$|f'(z)|\le1$. Intuitively, thinking of $f$ as a linear approximation of some analytic function within unit disc, we have $f(z)=az$, $a\in C$ for which (a) is necessary as $Range(f)\subseteq D(codomain)$ and (b) is necessary for $f(z)$ to be a contraction map.
A: The best interpretation of the Schwarz lemma I've heard of (I believe) comes from Eells and Sampson's article concerning harmonic maps. Indeed, let $f : M \to N$ be a holomorphic map between complex manifolds. The Schwarz lemma is a result of the maximum principle applied to the energy density $|\partial f |^2$. The energy density measures the extent to which $f(M)$ is stretched, i.e., how much tension (c.f., the notion of tension field for harmonic maps ;) ) when it sits inside $N$.
In particular, if $M$ is a curve of genus $g$ and $N$ is a curve of genus $g+1$, then $N$ is more hyperbolic than $M$. Hence, $M$ will be stretched when it's mapped to $N$ under $f$. Since holomorphic maps can't be stretched, $f$ will be constant.
