  We define the notion of special Lagrangian curvature, showing how it may be
interpreted as an alternative higher dimensional generalisation of two
dimensional Gaussian curvature. We obtain first a local rigidity result for
this curvature when the ambiant manifold has negative sectional curvature. We
then show how this curvature relates to the canonical special Legendrian
structure of spherical subbundles of the tangent bundle of the ambiant
manifold. This allows us to establish a strong compactness result. In the case
where the special Lagrangian angle equals $(n-1)\pi/2$, we obtain compactness
modulo a unique mode of degeneration, where a sequence of hypersurfaces wraps
ever tighter round a geodesic.
