Let $\mathbb{D}$ be the closed unit disk and let $u: \mathbb{D} \to \mathbb{R}^n$ be a smooth embedding. We say that $u$ is a harmonic map if $\operatorname{Tr}(\nabla d u)=0.$
Basic question: If $u$ is a harmonic map, can one deduce a priori bounds on the diameter of $u$ in terms of its boundary value?
Broader question: I am looking for a class of geometric equations on maps $u: \mathbb{D} \to \mathbb{R}^n$ whose solutions satisfy a priori estimates in terms of their boundary values. In particular, if I fix $C:= u(\partial \mathbb{D}),$ is there a class (or classes) of maps which would allow me to bound the diameter of $u$ in terms of $C$?
I know very little Riemannian geometry and realize that my question is both vague and open-ended. I would be grateful for any type of answer, including reading or reference suggestions.
EDIT: after reading Anthony Carapetis's comment, I realized that it was a mistake to state my question in terms of maps into $\mathbb{R}^n.$ I assumed that this would make the question simpler without fundamentally changing the answer, but the maximum principle provides an answer which seems unique to $\mathbb{R}^n.$ In fact, what I am really looking for is a class of maps $u: \mathbb{D}^2 \to N^n$ admitting a priori bounds on the diameter in terms of boundary data (e.g. the diameter of the boundary).