  We prove a quaternionic contact versions of the Obata's sphere theorems. We
show that if the first positive eigenvalue of the sub-Laplacian on a compact qc
manifold of dimension bigger than seven takes the smallest possible value then,
up to a homothety of the qc structure, the manifold is qc equivalent to the
standard 3-Sasakian sphere. We also give a version of the theorem on
non-compact qc manifold which is complete with respect to the associated
Riemannian metric using the existence of a function with traceless horizontal
Hessian. A qc version of the Liouville theorem is shown for qc-conformal maps
between open connected sets of the 3-Sasakian sphere.
