Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset
R^d \to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak
(1967) generalized this result and showed that if a sequence $f_n$
satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine
map.
In this talk I will discuss generalizations of these theorems to mappings
between manifolds and sketch the main ideas of the proof (using techniques
from the calculus of variations and from harmonic analysis).
Finally, I will describe how these rigidity questions are related to weak
notions of convergence of manifolds and present some open questions.
Based on a joint work with Asaf Shachar and Raz Kupferman.