Special Lagrangian curvature

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 ambient 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 ambient manifold. This allows us to establish a strong compactness result. In the case where the special Lagrangian angle equals (n − 1)π/2, we obtain compactness modulo a unique mode of degeneration, where a sequence of hypersurfaces wraps ever tighter round a geodesic.